Valverde was brought in as a replacement for the sacked Mauricio Pellegrino two weeks ago and has overseen victories away at Osasuna in the league and in the King's Cup, but his side offered little for the demanding Mestalla crowd.
After an evenly-balanced contest with few chances, home defender Tino Costa needlessly hauled down Roberto Trashorras in the area for a penalty in the 83rd minute.
Rayo's Alejandro Dominguez stepped up to score from the spot, and the home fans whistled and chanted against club president Manuel Llorente in the closing minutes.
Valencia's objective is a return to the Champions League, where they have already reached the last 16 this season, but they lie seven points adrift of fourth-placed Malaga in 11th place with 21 points.
City neighbours Levante were above them in fifth with 27 points after a 1-0 win at Real Zaragoza earlier in the day.
Juan Ignacio Martinez's side gained a measure of revenge for their defeat by the same scoreline at home to Zaragoza in a King's Cup last-16 first leg during the week.
Ruben Garcia gave the visitors the lead after 19 minutes and their notoriously solid defence stood firm thereafter.
Later on Sunday, leaders Barcelona take on second-placed Atletico Madrid at the Nou Camp, and third-placed Real Madrid host Espanyol. (Reporting by Mark Elkington, editing by Toby Davis)