Viktor Tsygankov's 76th minute winner on Tuesday night saw De Gea caught out of position as Spain suffered their first defeat since November 2018.
The visitors had dominated the game in Kiev, taking 21 shots to Ukraine's two, but were unable to take their chances before being hit on the counterattack.
The performance raised more questions about Spain's lack of a reliable No. 1 -- and a natural goalscorer at the other end of the pitch -- ahead of next summer's European Championships.
"You can always do more," Luis Enrique said in his postmatch news conference when asked about De Gea's responsibility for the goal. "But to blame [De Gea]... it's a bad habit, you [journalists] love doing it."
"We could all do more, me first of all. When you get a cake you share it, and when you get a plate of food that's going bad, you share that too. We all could have done more, but in a game like this, if you blame De Gea, let's finish up and go home."
De Gea, 29 -- who joined United from Atletico Madrid in 2011 -- has been persistently criticised in Spain for his displays with the national team, more so since a disastrous showing at the 2018 World Cup.
He has kept his place, in part, due to a lack of reliable alternatives. Kepa had been viewed as De Gea's successor after his high-profile €80 million move to Chelsea in 2018 but has since struggled in England.
Kepa's mistakes in Chelsea's opening Premier League games against Brighton and Liverpool saw the club sign Edouard Mendy from Rennes as a potential replacement.