Uruguay international Valverde was shown a straight red card for a tactical foul on Morata, who was set to shoot in a one-on-one with Thibaut Courtois in the 25th minute of extra time.
Atletico were unable to convert the free-kick or make the most of their numerical advantage in the closing stages, with Madrid going on to triumph 4-1 on penalties.
Valverde only received a one-game ban for the challenge, which he says was a calculated decision taken with no malice.
Asked about the incident, Valverde told ABC: "I don't know if it put me in the heart of Real Madrid fans, but I am not proud of committing a foul against an opponent.
"I don't know anyone who likes fouls. The ideal is to end a game without making any, but fouls are part of football.
"But what I would really not be proud of is not doing anything for my team in that, or any, passage of play.
"It was a clean foul, not done to injure, of course!"
Valverde has made 32 appearances in all competitions for Madrid in 2019, 23 of which have come as part of Zinedine Zidane's starting line-up.
However, the 21-year-old does not feel like he has firmly established himself as one of the team's leading players.
"I don't feel like I've experienced my confirmation [as a Madrid player]. I continue with the mentality of the first day I entered the biggest dressing room in the world," said Valverde.
"There are many players here who are stars, who have been confirmed for a long time and I know from them that, to feel like you have earned a place in the team, you have to toil a lot beforehand. I am just toiling. My goals are clear in my head: learn and give everything."
Asked if he would like to spend the rest of his career with Madrid, Valverde replied: "It's the dream. I don't want to be anywhere other than at Real Madrid.
"I would like nothing more than to be useful and necessary for my dream team."