Switzerland scored twice in the final 10 minutes in a thrilling 3-3 draw and, following a goalless extra time, prevailed in the shootout after goalkeeper Yann Sommer saved from Mbappe.
Mbappe said France were “incredibly sad” after exiting the tournament and that his miss would keep him awake at night.
“I’m sorry about this penalty. I wanted to help the team but I failed,” Mbappe wrote on Instagram.
“It’s going to be hard to sleep after this, but unfortunately these are the ups and downs of this sport that I love so much.
“The most important (thing) will be to get back up again even stronger for future commitments. Congratulations and good luck to the Swiss team.”
Switzerland face Spain in the quarter-finals on Friday.