Pressure has been building on Emery due to the club's erratic form and their city neighbours Levante can draw level on 47 points with a win at home to Osasuna later on Sunday.
Levante, the season's surprise packages, play at Valencia's Mestalla stadium in the local derby next weekend.
"Now after the defeat there will be those demanding solutions," Emery said at a news conference. "Decisions are taken on a day-to-day basis and those that are taken will come when they are ready.
"I have to stay firm. I have to keep my chin up so that nobody gives me a slap on the back of the head."
Saturday's defeat was their second in a row following last week's 2-1 home reverse to bottom side Real Zaragoza, when they also let slip a one-goal lead.
The result prompted many fans to wave white handkerchiefs, the traditional sign in Spain of discontent with the coach.
"The team is not in a good state and we have to recover," Emery said.
"Now I offer hard work and dedication. The supporters have to be behind the team and if they do that we will go forward."
Valencia play their Europa League quarter-final first leg at Dutch side AZ Alkmaar on March 29.