For the sake of putting in an answer,
it is absolutely, totally commonplace for Americans, while on holiday (in France, Japan, China, India, Tasmania, or anywhere),
to work on their laptop (i.e., for their ordinary home company in the US).
Your question is literally "Can?" you do this, the answer is an overwhelming "Yes".
As a curiosity, note that recently the India eVisa was changed so indeed, you can actually "do business" even, while in India.
(Note the checkboxes where you fill in one or more purposes for informational purposes; the final stamp in your passport when you arrive is just the same.)
So in fact if for some reason you had business dealings with an Indian company, when you're on holiday in India, it's 10000% ok to actually go to meetings, do paperwork - even with India companies - while you're there!
There is no issue whatsoever with checking your email, writing some code, on your laptop while you're on holiday somewhere.