Gautama Buddha Is NOT An Avatar of Lord Vishnu

Gautama Buddha is not part of the Dasavathar. Buddhism is not part of Hinduism. The Buddha is also not considered an avatar of Vishnu in Madhva (Dvaita), Smarta and Advaita traditions. The archaryas in the following videos have clearly shown the difference between the Buddha of Sanatan Dharma and Gautam Buddha of Buddha Dharma:

1. Their time of birth is considered different.
2. Both are from Gautam Gotra. Therefore, it is possible that "Gautam" is prepended before Buddha of Sanatan Dharm and hence the confusion.
3. There varnas are different. In Dashavatar, Buddha is considered a Brahmin while Gautam Buddha of Buddhism is Kshatriya.
4. There place of birth wide apart. Buddha of Sanatan Dharm took birth at Gaya, Bihar while Gautam Buddha is known to taken birth at Lumbini, Nepal.
5. Buddhism in its core does not believe in Bhagwan but focuses on method to achieve nirvana and also believes in Reincarnation. This is clearly against tenets of Vaishnavism which clearly considers Vishnu as Parameshwar.

Archaryas like Thirujnana Sambandar, Sakkiya Nayanar and Manivachaga Peruman have criticised the wrong and dreadful precepts in the Buddhism propounded by Gautama Buddha.

The Hari Vamsa Purana, the Visnu Purana, the Bhagavat Purana, the Garuḍa Purana, the Agni Purana, the Naradiya Purana, the Padma Purana, the Liṅga Purana, the Parasariya Maha Hora Sastra have shown the Buddha as one who had come to delude people, specially the asuras (demons), away from the correct teachings of the Veda so that these asuras would become less powerful than the Devas who followed the Vedas. (Listen to the Sankaracharya's satsang for details concerning this).

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]