No, Asoka is not a cakkavatti (ruler of the world), but dipacakkavatti (ruler of the continent, i.e Jambudipa or continent of west):
Asoka.-King of Magadha. He was the son of Bindusāra. Bindusāra had sixteen wives who bore him 101 sons.
The chief Pāli sources of information regarding Asoka are Dīpavamsa (chaps. i., v., vi., vii., xi., etc.), Mahāvamsa (v., xi., xx., etc.), Samantapāsādikā (pp. 35 ff. ). Other sources are the Divyāvadāna passim, and the Avadānasataka ii.200ff. For an exhaustive discussion of the sources and their contents see Prszlyski, La Legende de l'Empereur Asoka.
The Pāli Chronicles (Dīpavamsa and Mahāvamsa) mention only three of the sons, viz. Sumana (Susīma according to the northern legends) the eldest, Asoka, and Tissa (uterine brother of Asoka) the youngest. The Mahāvamsa Tīka (p.125; Mbv.98. In the northern tradition, e.g., Asokāvadānamālā, she is called Subhadrāngī, daughter of a brahmin of Campā) gives the name of his mother as Dhammā and calls her Aggamahesī (Bindusāra's chief queen); she belonged to the Moriyavamsa. The preceptor of Dhammā's family was an ājīvaka called Janasāna (which probably explains Asoka's earlier patronage of the ājīvakas).
In his youth Asoka was appointed Governor of Avanti with his capital at Ujjeni. The Divy. says he was in Takkasilā with headquarters in Uttarāpatha, where he superseded Susīma and quelled a rebellion. When Bindusāra lay on his death-bed, Asoka left Ujjeni and came to Pātalīputta where he made himself master of the city and possessor of the throne. He is stated in the Mahāvamsa (v.20; Mbv.98) to have killed all his brothers except Tissa that he might accomplish his purpose, and to have been called Candāsoka on account of this outrage (Mhv.v.189). It is impossible to say how much truth there is in this account of the accession. Asoka's Rock Edicts seem to indicate that he had numerous brothers, sisters and relations alive at the time they were written in Pātaliputta and other towns (see Mookherji, Asoka, pp. 3-6). His brother Tissa he appointed as his uparāja (advisor) (Mhv.v.33), but Tissa became a religious devotee attaining arahantship. The Theragāthā Commentary refers to another younger brother of Asoka, Vitasoka, who also became an arahant. (i.295f. The northern works give quite a different account of his brothers, see Mookherji, p.6).
Asoka had several wives. His first wife was the daughter of a merchant of Vedisagiri, whom he met when stopping at the merchant's house on his way to Ujjeni (Mhv.xiii.8ff). Her name was Devī, also called Vedisa-Mahādevī, and she was a Sākiyan, descended from a Sākiyan family who migrated to Vedisa to escape from Vidūdabha (Mbv., pp.98, 116). Of Devī were born a son Mahinda, and a daughter Sanghamittā, who became the wife of Aggibrahmā and mother of Sumana. Devī evidently did not follow Asoka to Pātaliputta, for his aggamahesī (chief-queen) there was Asandhamittā (Mhv.v.85). Asandhamittā died in the thirtieth year of Asoka's reign, and four years later he raised Tissarakkhā to the rank of queen. Mhv.xx.1-3. The Allahabad Pillar Inscription mentions another queen, Kāruvākī, mother of Tivara. The Divy. (chap. xxvii.) gives another, Padmāvatī, Kunāla's mother. Besides the children mentioned above, names of others are given: Jalauka, Cārumatī (Mookherji. p.9).
According to Mahāvamsa (v.21, 22), Asoka's accession was 218 years after the Buddha's death and his coronation was four years later. The chronicles (v.22ff) contain various stories of his miraculous powers. His command spread a yojana into the air and a yojana under the earth. The devas supplied him daily with water from the Anotatta Lake and with other luxuries from elsewhere. Yakkhas, Nāgas and even mice and karavīka birds ministered to his comfort, and thoughtful animals came and died outside his kitchen in order to provide him with food.
At first Asoka maintained the alms instituted by his father, but soon, being disappointed in the recipients, he began looking out for holy men. It was then that he saw from his window, his nephew, the young novice Nigrodha. Owing to their friendship in a past birth [Asoka, Devanampiyatissa and Nigrodha had been brothers, traders in honey, and they gave honey to a Pacceka Buddha. Asandhamittā had been the maiden who showed the honey-shop to the Pacceka Buddha. The story is given in Mhv.v.49ff], Asoka was at once drawn to him and invited him into the palace. Nigrodha preached to him the Appamādavagga and the king was greatly pleased. He ceased his benefactions to other religious orders and transferred his patronage to Nigrodha and members of the Buddhist Order. His wealth, which, according to the Samantapāsādikā (i.52), amounted to 500,000 pieces daily, he now spent in doing acts of piety - giving 100,000 to Nigrodha to be used in any manner he wished, a like sum for the offering of perfumes and flowers at the Buddha's shrines, 100,000 for the preaching of the Dhamma, 100,000 for the provision of comforts for members of the Order, and the remainder for medicines for the sick. To Nigrodha, in addition to other gifts, he sent sets of robes three times each day, placing them on the back of an elephant, adorned by festoons of flowers. Nigrodha gave these robes to other monks (MA.ii.931).
Having learnt from Moggaliputta-Tissa that there were 84,000 sections of the Dhamma, he built in various towns an equal number of vihāras, and in Pātaliputta he erected the Asokārāma. With the aid of the Nāga king Mahākāla, he created a life-size figure of the Buddha, to which he made great offerings.
His two children, Mahinda and Sanghamittā, aged respectively twenty and eighteen, he ordained under Moggaliputta-Tissa and Dhammapālā, in the sixth year of his reign (MA.v.197, 209). This raised him from a paccadāyaka to a sāsanadāyādin.
In order to purge the Order of undesirable monks and heretical doctrines, Moggaliputta-Tissa held the Third Council under the king's patronage. It is said that the pious monks refused to hold the uposatha with those they considered unworthy. The king, desirous of bringing about unity in the Sangha, sent a minister to restore amity, but the minister, misunderstanding his orders, beheaded many holy monks, being at last stopped by the king's brother Tissa, who was then a monk (MA.vs.240ff).
At the conclusion of the Council, held in the seventeenth year of his reign (Ibid., 280; in the northern texts Moggaliputta-Tissa's name is given as Upagupta. It was for this Council that the Kathāvatthu was written), Asoka sent forth Theras to propagate the Buddha's religion:
* Majjhantika to Kasmīra and Gandhāra,
* Mahādeva to Mahisamandala,
* Rakkhita to Vanavāsa,
* Yona Dhammarakkhita to Aparantaka,
* Mahārakkhita to Yona,
* Majjhima to the Himālaya country and
* Sona and Uttara to Suvannabhūmi;
* Mahinda with Itthiya, Uttiya, Sambala and Bhaddasāla he sent to Lankā
(Ibid., xii.1-8. For particulars of these missions and identification of the places mentioned, see under the different names; this list appears also in the Samantapāsādikā, where further interesting details are given. For a discussion on them see Mookherji, pp.33ff). In the eighteenth year of his reign he sent to Lankā, at Devanampiyatissa's request, Sanghamittā, with a branch of the great Bodhi Tree at Buddhagayā (Mhv.xx.1). A little earlier he had sent by his grandson Sumana, some relics of the Buddha and the Buddha's alms-bowl to be deposited in the thūpas of Lankā (Mhv.xvii.10f).
Asoka reigned for thirty-seven years (Mhv.xx.6). In his later life he came to be called Dhammāsoka on account of his pious deeds (Mhv.v.189). The Dīpavamsa gives his name in several places as Piyadassī. E.g., vi.1, 2, 25. The title Devānampiya used by Asoka in his inscriptions was also used by Tissa, Asoka's contemporary in Ceylon, and by Asoka's grandson Dasaratha (Nāgarjunī Hill Cave Inscription). It was used also by other kings in Ceylon: Vankanāsika Tissa, Gajabāhukagāminī and Mahallaka-Nāga (Ep. Zeyl. i.60.f).
The Chronicles state that Asoka and Devanampiya Tissa of Ceylon had been friends - though they had never seen each other - even before Mahinda's mission to Ceylon. Tissa had sent him, as a friendly gesture, various gifts, and Asoka had returned the courtesy. He sent an embassy of his chosen ministers, bearing gifts marvellous in splendour, that Tissa might go through a second coronation ceremony, and the messengers were directed to give this special message to the king: "I have taken refuge in the Buddha, Dhamma and Sangha and declared myself a follower of the religion of the Sākyaputta. Seek then, even thou, oh best of men, converting thy mind with believing heart, refuge in these best of gems." (Mhv.xi.18-36)
The Milindapanha (p.121) mentions an encounter of Asoka with a courtesan of Pātalīputta, Bindumatī, who, in order to show the king the power of an Act of Truth, made the waters of the Ganges to flow back.
According to the Petavatthu Atthakathā (244ff) there was a king of Surattha, called Pingala, who used to visit Asoka in order to give him counsel. Perhaps he was an old friend or tutor of the king.
Asoka is called a dīpacakkavatti as opposed to padesarājās like Bimbisāra and Pasenadi (Sp.ii.309).
Asoka had three palaces for the three seasons: Mahāsappika, Moragīva and Mangala (Ras.i.93).