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.”
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Kepa, the club’s record £71m signing, defied Sarri’s attempt to substitute him for Caballero before Manchester City won on penalties. The Italian appeared furious and walked down the tunnel before quickly returning.
The 24-year-old former Athletic Bilbao player said: “I wanted to take the time today to apologies fully and in person to the coach, to Willy, my team-mates and to the club.
“I have done this and now I want to offer the same apology to the fans. I will learn from this episode and will accept any punishment or discipline the club decides is appropriate.”
The club will donate Kepa’s fine to the Chelsea Foundation.