Currently, you can invest up to USD 250,000 every year in foreign stocks from India. This amount can change, subject to RBI guidelines. So your investments in US securities are also governed by the same limit.
Foreign investments fall under clearly defined RBI guidelines. The remittance of money for foreign investments comes under the Liberalized Remittance Scheme (LRS).
Under LRS an Individual can remit up to USD 250,000 per financial year to invest in foreign equities done through an authorized dealer (commonly, your bank). As per RBI policy, having a PAN card is required to purchase shares in foreign countries.