Foreign Minister SM Krishna described Monday's death as "unfortunate" and said the matter had been taken up with the United States.
Three Indian fishermen were also wounded in the firing incident.
The fishermen were in a small boat off the coast of Dubai, when the US navy ship fired on them on Monday.
The dead man was from the southern Indian state of Tamil Nadu, as are his wounded colleagues.
"Our embassy in Abu Dhabi is working with local authorities to carry out a full investigation into the circumstances leading to this unfortunate and tragic incident. The acting consul general is in Jebel Ali to provide all necessary assistance," the Indian foreign ministry said in a statement.
One of the injured fishermen, speaking to Reuters news agency from hospital in Dubai, said they had not been aware of any warnings.
"We had no warning at all from the ship, we were speeding up to try and go around them and then suddenly we got fired at," Muthu Muniraj, 28, said.
He was shot in the legs during the incident.
"We know warning signs and sounds and there were none. It was very sudden. My friend was killed, he's gone. I don't understand what happened."