Sidwell will miss Fulham's last four games of the season after he saw red for the second time in two appearances.
The former Arsenal midfielder, who was given a straight red for a dangerous tackle on QPR defender Armand Traore three weeks ago, caught Mikel Arteta on his standing foot with a two-footed lunge just 12 minutes in to the London derby at Craven Cottage.
Jol said: "I can't defend him because that would only sound ridiculous. I was annoyed with him but he says it is not intentional. He will regret that tackle. It was the first foul that spoilt the game.
"I know it was not intentional because he wanted to play. He has just come back from a three-match suspension and now he is out for another four matches again so it was not only disappointing him, but also for us because we had to play with 10 men for a long time so that was more disappointing."
Despite the setback of losing Sidwell so early in the game, Fulham coped admirably, continuing to probe the Arsenal back four thanks to impressive performances from Urby Emanuelson, Eyong Enoh and Alex Kacaniklic.
They were left exposed at the back, however, and Per Mertesacker took advantage of some slack defending from a free-kick to lash home just before half-time.
Jol, whose team are nine points above the drop zone, criticised Bryan Ruiz for failing to stop Mertesacker scoring from Laurent Koscielny's knock-down.
He said: "Bryan Ruiz is my sixth man at the end of the wall and he was not doing what he normally should do - stay on the inside of the player - so that was disappointing."
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- Armand Traore