La Liga champions Barca could only manage a point at Valencia as their winless streak in the league extended to four matches on Sunday.
Barca centre-back Pique – who made an error in last month's shock 2-1 loss to Leganes – was sloppy in his clearance that led to Ezequiel Garay's second-minute opener.
Barcelona recovered to dominate much of the run of play with Lionel Messi equalizing midway through the first half , but the Catalans could not find a winner and walked away with just a share of the spoils.
The draw still moved Ernesto Valverde's side up to second place in the league, ahead of both Atletico and Real Madrid and just a point behind suprise leaders Sevilla, and Pique was not keen to hear from the critics after the match.
"I know that there are many people who are hoping that things don't go well for me," he told reporters.
"They can enjoy it, they should come out of their caves and enjoy the sun. There are many people waiting for me and want to go for me.
"When we concede a goal it's my fault, there are people who are waiting for me, but not only in Madrid.
"The TV can take advantage, come out of the cave, because this is going to change."
Barcelona will get the chance to go top of the table following the international break as they square off with Sevilla in a table-topping clash.
The club's fortunes have been better in the Champions League where they have won two out of two, including a 4-2 victory over Tottenham midweek, and Pique believes the style of play in Spain means La Liga opponents are often better prepared to slow his side down.
"Tottenham style helped us at Wembley, Arthur and our midfielders were able to keep the ball and in Spain all teams use to get closed vs us and it gets harder to break the two lines of four players inside their area," he said. "We're used to passing the ball to open up spaces, but it is not always possible."