Gautama Buddha Is NOT An Avatar of Lord Vishnu

Gautama Buddha is not part of the Daśāvatāra. Buddhism is not part of Hinduism. The Buddha is also not considered an avatāra of Viṣṇu in the Madhva (Dvaita), Smārta, and Advaita traditions. The āchāryas in the following videos have clearly shown the difference between the Buddha of Sanātana Dharma and Gautama Buddha of Bauddha Dharma:

1. Their times of birth are considered different.
2. Both are from the Gautama gotra; therefore, it is possible that "Gautama" was prefixed to the Buddha of Sanātana Dharma, thus giving rise to the confusion.
3. Their varṇas are different. In the Daśāvatāra, the Buddha is considered a Brāhmaṇa, whereas Gautama Buddha of Buddhism was a Kṣatriya.
4. Their places of birth are widely separated. The Buddha of Sanātana Dharma is believed to have been born in Gayā, Bihar, whereas Gautama Buddha is known to have been born in Lumbinī, Nepal.
5. Buddhism, at its core, does not believe in Bhagavān, but instead focuses on methods to achieve nirvāṇa, and also holds a belief in reincarnation. This stands in clear contrast to the tenets of Vaiṣṇavism, which regards Viṣṇu as the Parameśvara.

Āchāryas such as Thirujñāna Sambandhar, Śakkiya Nāyanār, and Maṇivācaka Perumāṉ have criticised the erroneous and pernicious doctrines found in the form of Buddhism propounded by Gautama Buddha.

The Harivaṃśa Purāṇa, the Viṣṇu Purāṇa, the Bhāgavata Purāṇa, the Garuḍa Purāṇa, the Agni Purāṇa, the Nāradiya Purāṇa, the Padma Purāṇa, the Liṅga Purāṇa, and the Parāśarīya Mahā Horā Śāstra depict the Buddha as one who appeared to delude people—particularly the Asuras (demons)—and divert them from the true teachings of the Veda, so that these Asuras would become less powerful than the Devas, who remained adherents of the Vedas. (Refer to the Śaṅkarāchārya's satsaṅga for further details.)

The Agni Purana [16/1-3] and Vishnu Purana [18/13-18] refer to Buddha as an embodiment of Grand Deception (mayamohasvarupa) and that he deluded the people from the Vedic Religion. 

In ‘the Ayodhya Kanda of Valmiki Ramayana, the Buddha has been compared to a thief and the Tathagata to atheists (nāstika). It says:

yathā hi corasya tathā hi buddhastathāgataṃ nāstikaṃatra vidhiḥ ||

(Meaning: As a thief, so is a Buddha. Know that Tathāgatas are atheists.) [Sarga 109; śloka 34]

The Yajnavalkya smriti says that the very sight of a monk with red robes and shaven head (referring to Buddhists), even in a dream, is a bad sign. [Yajnavalkya smriti /272-273].

Adi Sankaracarya has said:

And on account of its general deficiency in probability.

No further special discussion is in fact required. From whatever new points of view the Bauddha system is tested with reference to its probability, it gives way on all sides, like the walls of a well dug in sandy soil. It has, in fact, no foundation whatever to rest upon, and hence the attempts to use it as a guide in the practical concerns of life are mere folly.--Moreover, Buddha by propounding the three mutually contradictory systems, teaching respectively the reality of the external world, the reality of ideas only, and general nothingness, has himself made it clear either that he was a man given to make incoherent assertions, or else that hatred of all beings induced him to propound absurd doctrines by accepting which they would become thoroughly confused.--So that--and this the Sūtra means to indicate--Buddha's doctrine has to be entirely disregarded by all those who have a regard for their own happiness. [Adi Sankara Bhasya on Brahmasutra 2/2/32]