The Spaniard refused to be replaced by Willy Caballero at Wembley.
“Although there was a misunderstanding, on reflection, I made a big mistake with how I handled the situation,” Kepa said in a Chelsea statement.
Manager Maurizio Sarri said he and Kepa had since had “a good conversation”.
Sarri, who reacted angrily when Kepa refused to leave the field towards the end of extra time, also said the incident had been “a misunderstanding”.
But he added:
“Kepa realises he made a big mistake in the way he reacted.
“He has apologised to me, his team-mates and the club. It is up to the club if they want to discipline him according to the club rules, but for me this matter is now closed.
“The team performance as a whole was extremely positive and it is a shame to see how this incident has overshadowed our efforts in what was a very competitive cup final.”