Are US employers responsible for any taxes in India if they hire the services of a US Independent Contractor residing in India?

No, they are not.

I generally agree with ATE (though this is really an Indian tax question that should be posed to an Indian tax advisor). Of course, if the "independent contractors" are really "employees" of the "US employers" then some Indian taxes may be due.

1. You will work in India. So this will be considered India source income and is taxable in India.
2. You are U.S. citizen and are being paid by the U.S. employer, so the employer will withhold U.S. taxes from your income (if you are an employee). If you are an independent contractor, then the employer may not hold taxes. But your income is subject to SE tax at 15.3% plus income taxes in the U.S. Even your State may like to tax you.
3. So your income will be double taxes. One of the countries will provide you credit for taxes paid to the other country. Which country?