Rama was aware he was an incarnation of Vishnu, and importantly he also prayed to and was a great devotee of Shiva, leading to a linga temple rising in Ramaeswaram. This is is complete contrast to Abraham, who would have denied Rama as Vishnu incarnate, and firmly opposed the linga idol and Shiva as divine. In this way he continues the animosity of Brahma against Shiva, but as a man who submits to the One God.
This only shows that Rama follows a morality of keeping an oath to honour his father and to not act in self-interest. In Judaism the making of oaths should be avoided, and we also see Abram acting in self-interest multiple times throughout Genesis.
The problem is, whilst a follower of the Vedic scriptures will say both Rama and Abraham worship One God, a follower of the Hebrew scripture will never agree with it. It is clear in Genesis that Abraham only worships the One God who is also without form or name (at that point in time). The existence of a temple with an idol to the diety Shiva as a symbol of Rama's devotion shows that, even if Rama's intent is to worship one God, or Brahman, this is incompatible with how Abraham shows his devotion, primarily to submission and fear of, his One God.
In fact Judaism makes determined efforts to reject the Sanskrit scriptures in general, and Shiva in particular. For example, in Vedic mythology Brahma the Creator God generates a consort Saraswati so that he can further the purpose of creation. However, he begins to lust after her intensely even when she tries to escape, growing a fifth head to catch her. Shiva becomes enraged that Brahma would lust after his own daughter and cuts off his fifth head as punishment.
I would argue that the covenent of circumcision, after God makes Abram into Abraham, is in testament to holding One God and supremacy without equal. As Shiva is represented by the linga, or phallus, the act of circumcision to purposefully remove the skin covering the head of the penis (foreskin) is a powerful symbol of conquering Shiva and rejecting the Sanskrit teachings, as well as being a physical and irreversible sign of submission to Abraham's God. Circumcision would be seen as a very committed act of devotion also when viewed through the lens of an outsider, particularly as it is performed when a baby boy is 8 days old.