Why does the UK have carriers? Or France? Neither intends nor expects to fight the USA. But they might want to act independently, in an operation where carriers are useful, or even essential (think Falklands). India is the same.
The Indian ocean is surrounded by fragile, & in some cases failed, states. One can imagine scenarios in which India might want to - or be asked to - intervene. India is also building up her amphibious forces, building new landing ships (based on the old British Round Tables) & buying a second hand LPD from the USA for now, but there's talk of more in the future. Carriers which can function as either primarily fixed-wing, or primarily helicopter carriers, depending on task, are very useful for supporting amphibious operations, even where the opposition doesn't have an air force.
None of this has anything to do with the USA. Indian carriers could sail straight past US carriers, en route to (e.g. - don't read anything into it) support a landing on Mauritius, & exchange polite salutes.
BTW, I hope they get a move on in Cochin. I want to see that ship launched!