In his first singles match since January, Gasquet was the better man as he defeated Murray 6-4 6-4.
Gasquet has had his own injury problems in recent years and struggled to regain the form which saw him go as high as seventh in the world.
Murray will not play singles at the US Open this year, instead opting to play in the doubles.
But he will resume singles activity in the Autumn when he competes at the Zhuhai Championships and China Open.
He could then receive a wildcard into the Shanghai Masters and indoors tournaments in Vienna and Basel.
Speaking after his win, Gasquet was asked whether he had any words for Murray but the Frenchman was surprisingly hesitant.
“You know, I have nothing to say to him. He’s a much better player than me,” he said.
“He won Grand Slam. I only did semifinal. He has many things to tell to me. Him, I have really nothing to say to him.
“He’s a better player than me, so I just hope he, step after step, day after day, tournament after tournament, he will climb, you know, the matches, the practice, and he will come back strong as before.”
However, Gasquet was more forthcoming about what he saw from Murray and the difference compared to previous matches.
“Of course he was the best player physically with — he was everywhere, playing incredible. So, of course, it’s the first match for him. He didn’t play for seven months, so it’s very difficult, you know,” he said.
“He’s only at the beginning. You know, he has a lot of steps to come back as before. But I think he can, feeling honestly, he will do it. I’m sure about it.
“He’s still one of the best players. He’s playing great. The return, as I said, the serve, he’s still one of the best in the world.
“I think just physically, you know, need to move. Of course it’s normal. It’s his first match. Before, you know, impossible to do a winner against him.
“Now it’s a little bit different. It’s normal because it’s his first match.”