Frustrated by his injury and Gasquet's near flawless form, Kyrgios raged at his supporters to "get out of my box, I'm sick of it. Get out of my box."
His brother Christos and members of his fitness staff were among those who wore Kyrgios' frustrations.
Asked about the comments post-match, he said: "Don't want to talk about it."
Kyrgios was jeered and whistled at after swatting his courtside bench with his racquet, an action which escaped penalty.
The world No 19 is among a host of players to feel the wrath of French officials determined to eradicate rebellious behaviour.
World No 2 Andy Murray ($4650) and Jack Sock ($3900) were also penalised for audible obscenities.
Benoit Paire, who was fined $4650 for unsportsmanlike behaviour, and Victoria Azarenka, $3100 for racquet abuse, also were singled out.
Kyrgios is confident he will recover from shoulder soreness ahead of Wimbledon.
Kyrgios was clearly hampered by the injury, but was reluctant to use the problem as an excuse after being "absolutely destroyed" in straight sets by Richard Gasquet.
Kyrgios will now rest and recuperate ahead of the Queen's Club lead-up to Wimbledon.
"I don't have another tournament for a while now," he said
"I can just get treatment and just look after it. Yeah.
"I felt it this morning when I was in my warm-up, and I didn't think it was anything, but it was bothering me a little bit and I got physio on it.
"It actually loosened up toward the start of the second set, so it's hopefully not too bad. Feels okay now."
Kyrgios cut a desolate figure post-match after being taken apart by ninth seed Gasquet.
"I got absolutely destroyed. Wasn't really fun," Kyrgios said.
"I struggled today pretty much in every aspect. I had chances in the second and I just didn't play big. I just played really bad today. Really bad.
"Leaves a bit of bad taste in my mouth. I thought I had a good clay court season leading up, and it was just a bad performance today."
Kyrgios confirmed the injury impacted his backhand and ball toss.
