U.S. LLCs can generally hold other entities but first require a tax election to not be treated as a disregarded entity. LLCs are treated as a disregarded entity by default if there is only one owner. To change this, an election must be filed with the IRS to turn it back into an LLC or even a corporation for tax purposes. Once this is done, the LLC can generally own whatever it wants.
On the India law side, there can be restrictions on foreigners holding stock in an Indian company. A U.S. LLC is probably considered a foreigner for purposes of this test. Once these restrictions are cleared up, then the transactions is likely possible.