Henry, playing in his second match since rejoining Arsenal on loan earlier this month, admitted losing his temper when an Arsenal supporter insulted him at Liberty Stadium.
"I was really shocked and disappointed on Sunday, and not just at the result," said Henry.
"After the final whistle of a very tricky game, I insisted the whole team go and applaud the Arsenal supporters.
"When I arrived in front of our own fans, I couldn't understand the abuse the team were getting from one of them.
"I wanted to tell the fan in question that we need their support not abuse. The sentiment was right but in hindsight could have been better expressed.
"So I would like to first and foremost apologise for using inappropriate language to that particular 'supporter'.
"I didn't intend to be malicious or threatening and am aware that it could have been said in a calmer manner."
The result left Arsenal in fifth place and following leaders Manchester City's 1-0 win at Wigan Athletic on Monday, they are now 15 points behind City and four points adrift of Chelsea in fourth place.
Arsenal face second-placed Manchester United at home on Sunday and in a plea to fans, Henry added: "We need your support in the good times, but more importantly the bad.
"Get behind the team against Manchester United on Sunday and try to get three points to get the season back on track."
Henry, Arsenal's all-time record scorer, fired the winning goal against Leeds United in the FA Cup last week when he came on as a second half substitute, but he failed to find the target at Swansea.
He is due to return to his New York Red Bulls team at the end of next month before the start of the new Major League Soccer season.
