The Briton returned to action at the Cincinnati Masters after being sidelined since January with a career-threatening hip injury.
Following the after the 6-4, 6-4 defeat, Murray announced that he will not be playing in the US Open singles later this month.
"There were a lot of things I would like to have done better in the match, but you also have to be somewhat realistic in terms of what you can expect, in terms of how you actually play and hit the ball," Murray said.
"My legs were a little bit heavy at the end of the match. When Richard drop-shotted me, there were a few times I didn't even run to the ball, didn't react to it.”
The three-time grand slam champion, now ranked 324th, hit a respectable 20 winners but struck 28 unforced errors.
Murray hadn’t played a singles match since a first-round exit at the Australian Open, where he tearfully acknowledged he feared his career was over.
Since undergoing hip surgery in January, the 32-year-old has slowly fashioned a return to the court.
He has played doubles for the past two months, winning at Queen's Club in London with Feliciano Lopez.
Gasquet, however, who won the Cincinnati Masters in 2008 and 2011, was too strong for his opponent.
Despite the defeat, Murray remains optimistic about the future and improving further.
"I think with a few more weeks of practice and hopefully some more matches, then I could do better."