Can an Indian citizen on H-1B travel to visa-free countries for Indians (such as Nepal, Thailand) without an H-1B stamped on an Indian passport?

1. The US airport you leave from won't mind if you check in with your Indian passport to travel to a country where Indian passport holders do not require a visa. As far as the checkin staff are concerned, you may not even be coming back to the US afterward.

2. The third-country airport you leave from in order to return to the US may want to verify that you have the right to arrive in the US and not be turned away. You will quite likely need to provide some proof that you have a US visa. Do you have a way to prove that? If you can't prove you will be able to successfully enter the US, the airline may refuse you because they don't want to have to fly you back out of the US if you can't enter. I would think that whatever proof you would show to the US authorities on entry would be sufficient to show at the airport you're leaving from, but if that proof is entirely electronic and only visible in US computer systems, you may have a problem.

The H1-B or any other US visa stamp is needed to enter the country. You do not need this to exit the US and in many cases you do not need this when you are in the US. But you cannot enter the US (or board a plane to the US) without one (assuming you are not a US citizen or permanent resident).

When you visit a third country they may expect a return ticket in addition to validating you are allowed to enter said country to make sure you exit the country in a timely fashion. If your "return" ticket is to India, there are no additional complications. If your "return" ticket is to the US they may not allow you to enter as you do not have a way to go back to the US (no valid US visa)*. Please look up return ticket requirements for the said country. If you are denied entry, you may end up spending some time in a detention facility while they figure out how to get you to India, which may not be very pleasant.

The bigger question is what is your plan. If you intend to travel to, say, Sri Lanka, and get back to the US, you need to plan to get a visa stamp and figure out where you should get the stamping from. If your plan is to visit the country on the way to India and then return to the US, you should plan to a get a visa stamp in India before returning to the US.

*It is also not uncommon for Indian citizens to get their US visa stamped from a third country in certain specific situations but this does not seem to be your plan.