US doesn't have exit controls on their borders, so it's not a problem leaving the US even if you're out of status in America.
The best course of action for most folks in such a scenario is to file a I-539 change of status before their status expires to transition from H1-B into B1/B2 visitor status.
Another potential issue is if you transit through a third country which may require a valid visa for the country that you started your journey from. This is definitely not needed for transiting through the EU or the Middle East but you should check if the transit requirements if they apply in your case.
You've said that you're flying direct from USA to India (with a possible hop within the US), there is no issue with regards to a transit here. Furthermore when you arrive in India, they have no idea what your status was in the USA or when it expired, plenty of people return back with expired visa stamps but valid I-797s for visa stamping and from the Indian customs side your case is indistinguishable from those. That said, it really doesn't matter, overstaying in a foreign country doesn't impact your ability to enter India as an Indian citizen with a Indian passport.
You will need to declare this overstay in future visa applications both to the USA and to any other country which might ask this question, whether 1 week of overstay will matter in future visa issuals or not I can't say.