There are two main sources for foreign currency reserves: a) export earnings of Indian companies b) foreign investment coming into India.

When these dollars come in, they are selling dollars and buying INR to invest in India. This will cause the INR to appreciate in value. To prevent it from appreciating too much the Reserve Bank of India supplies INR in exchange for USD in the foreign exchange market. Thus, the RBI ends up accumulating reserves as a result of this policy.

The USD is a free-floating currency, whose value is set by FX markets. The government does not intervene in the FX markets. So, the US does not accumulate reserves. Another point, indirectly related to your question is this: Being the world's major reserve currency, the US borrows in its own currency, which it can always print.