Immigration and naturalization is pretty far out of my comfort zone, but I'm confident that the answer is yes.
Although people often believe that a foreign embassy is considered the territory of that country, I don't know of any law that supports that belief. Instead, through the Vienna Convention, the embassy grounds remain the territory of the host state but are provided a variety of protections and immunities because of their diplomatic status. With the embassy on U.S. soil, the child would therefore satisfy the "born ... in the United States" prong of the 14th Amendment's Citizenship Clause.*
But that would not be the end of the analysis, as birthright citizenship also requires not just that the child is born in the United States, but also that the child be "subject to the jurisdiction thereof."
So if the child were born to an American citizen who had entered the Indian embassy to get a travel visa, the child would be both born in the United States and subject to its jurisdiction, making it eligible for birthright citizenship. But if the child were born to Indian ambassador or to diplomatic staff, who would generally be able to claim diplomatic immunity, that child would not be subject to the jurisdiction of the United States and would not be able to claim birthright citizenship. Slaughter-House Cases, 83 U.S. 36, 73, (1872) ("The phrase, ‘subject to its jurisdiction’ was intended to exclude from its operation children of ministers, consuls, and citizens or subjects of foreign States born within the United States.").
* I haven't been able to find any cases saying this explicitly, but all the cases involving children born in foreign embassies sort of skip over the question as though they just assume that the child was born in the United States. See, e.g., Raya v. Clinton, 703 F. Supp. 2d 569 (W.D. Va. 2010); Nikoi v. Attorney Gen. of U.S., 939 F.2d 1065 (D.C. Cir. 1991) These cases also go on to conclude that those children are not citizens of the United States, because they are not "subject to the jurisdiction thereof."