Barcelona, seeking its third straight league title, is one point behind Madrid after 24 rounds. Madrid has a game in hand, at Celta Vigo.
The stories posted by Pique mention a goal he scored but was mistakenly disallowed for offside, and a Ramos goal that was not annulled even though he was offside.
Ramos said the accusation was expected from the outspoken Pique.
“It would surprise me if it came from (Lionel) Messi,” Ramos said. “But from Pique, we already know him, we won't get caught up in that. We will mind our own business, which is to win every match and secure the league title.
“Everyone has to stick to their own problems. Fortunately we only depend on ourselves. We are focused on improving and on doing what we have to do.”
It's not the first time Pique publicly showed his frustration with the referees in Spain.
Last month, after a penalty kick not awarded to Barcelona against Villarreal, he pointed his finger at the VIP box and yelled toward officials before leaving the field, apparently targeting Spanish league president Javier Tebas. Pique also complained after Barcelona's loss at Athletic Bilbao in the Copa del Rey.
The most controversial play on Sunday was the penalty call that led to Cristiano Ronaldo's equalizer in Villarreal. The ball hit the hand of Bruno Soriano inside the area, but the midfielder claimed it was involuntary.
“I can't just cut my arm off,” Soriano said.
Villarreal president Fernando Roig added fuel to the controversy by complaining the officiating crew left the stadium carrying Real Madrid bags. Madrid, like many other clubs, is known to hand out small gifts such as key chains, pins and pens to the referees and their assistants after matches.
“The referees try to be fair and sometimes they'll benefit you and sometimes they'll hurt you,” Ramos said.
Madrid is trying to win its first league title since 2012.
Barcelona, which has lifted three league trophies since then, trailed Madrid by as many as six points this season.
Third-place Sevilla, seeking its first league title since 1946, is three points off the lead. Fourth-place Atletico, which last won the league in 2014, is 10 points behind its city rival.
“Whoever thinks that Barcelona will throw in the towel and that Madrid has already won it with nine or 10 rounds to go, doesn't know a lot of football,” Ramos said. “It's going to be very difficult.”