Valencia had started the day in fourth place and in prime position to take the Champions League play-off place that came with the position, but their defeat and Real Sociedad's 1-0 win at Deportivo La Coruna saw them slip down to fifth.
Valverde was brought in to replace Mauricio Pellegrino in December on a six-month contract and last week put off a planned meeting to discuss his future until after the weekend.
"This has been my last game," Valverde told a news conference at the Sanchez Pizjuan.
"I am not very pleased, not just with the game and failing to get into the Champions League, but with the general situation at the club.
"There is instability with the resignations in the last couple of months. Without those I would have stayed. So I have made the decision I think had to be taken."
Financially-troubled Valencia have suffered an institutional crisis over the last few months, which led to president Manuel Llorente quitting. A replacement is set to be appointed next week.
"When I arrived at Valencia in December there was already fair amount of instability in the club, and the fans were calling for the president to go," Valverde added.
"There was a lot of unrest in my first few games too, but we worked hard to try and improve situation, to get rid of controversy and to keep the group united.
"If you ask why I waited so long to say this, as some people have said I was showing a lack of respect to the club, I did it to try and maintain the competitive spirit of the team."
Real Valladolid's Serbian coach Miroslav Djukic, who has said he will leave at the end of the season, is favourite to return to a club where he spent six years as a player and replace Valverde.
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