No foreign embassy can organize a tour of the White House. The White House staff organizes tours. But applications have to made either via a member of the Congress (in case of US citizens) or via a respective embassy in the Washington D.C. (in case of foreign nationals).
UPDATE: I've found several sources mentioning that embassies in DC (not just the Indian one) are indeed unwilling to apply for White House tours on behalf of their citizens. The only remaining option is to contact a member of the Congress. Foreigners of course have no MoC representing them, but there are reportedly some that accept requests from foreigners as well. One of them should be Eleanor Norton.
In any case, your odds are very low, since the demand far exceeds available slots. Even US citizens have it very hard to get a WH tour ticket.