De Gea conceded four goals at Goodison Park on Sunday.
He posted to social media: "Difficult to put what I am feeling today into words. As captain I want to say that the performance of the team today, on top of the result, was not up to the standard expected when you pull on this shirt.
"We know that we have to improve but words mean little: we have to give everything for this club.
"Thank you for your unconditional support."