The behaviour of the Portuguese international, who was playing in an unfamiliar central midfield role, was "shameful" and "intolerable," Marca wrote in an opinion piece, while As columnist Luis Nieto said he deserved "general condemnation" and should be punished.
Television replays showed that Pepe, already booked for a late tackle on Sergio Busquets, appeared to deliberately tread on Messi's hand about 20 minutes from the end of Barca's 2-1 win at the Bernabeu in an incident apparently missed by the referee.
His theatrical reaction after tangling with Cesc Fabregas earlier in the second half, when he rolled around clutching his face, also drew criticism.
"Violent, with excessive aggression, play-acting and a long way from what the behaviour of a footballer in a high-level match should be," Marca wrote.
"What's more, his problem is that he is a repeat offender," the paper added, recalling an incident during the 2008/09 season when Pepe kicked Getafe's Francisco Casquero while he was on the ground, an action that earned the Portuguese a 10-match ban.
"The Casquero episode cannot now be seen as an isolated one but instead must be viewed as the sad reality of a player who goes beyond over-excited.
"Pepe might be an excellent person away from the pitch - and he is - but someone urgently needs to make him reflect and help him to correct these shameful actions."
INVOLUNTARY ACT
The 28-year-old published a brief statement on the club website later on Thursday saying he had not meant to stamp on Messi and it had been unintentional.
"I nevertheless want to apologise to him if he was offended because all I do is defend my team and this institution," he added. "I give my heart and soul and the thought of hurting a colleague has never entered my mind."
Barca centre-back Gerard Pique told a news conference it was up to the football authorities to intervene.
"If they believe that the action went beyond the limit of what should happen on the pitch they are the ones who have to take it forward," the Spain international said.
"I don't believe that we as a club should be going and asking for a player to be punished."
A spokesman for the Spanish football federation (RFEF), which organises the King's Cup, said they had no comment.
The outcry over Pepe's behaviour deflected some of the attention away from Real's latest failure against Barca, who host their arch-rivals in the second leg on Wednesday.
The Spanish and European champions' victory was coach Pep Guardiola's ninth win in 13 'Clasicos' since he took the helm in 2008 and came on his 41st birthday.
His only reverse was in last year's King's Cup final, when Cristiano Ronaldo netted an extra-time winner in a 1-0 victory.
FAILED TACTIC
Real coach Jose Mourinho, who told a news conference Pepe should be punished if the stamp on Messi was deliberate, was also subjected to severe criticism in Marca and As.
The former Inter Milan and Chelsea manager's tactic of allowing Barca to dominate possession and rely on swift counter-attacks had proved a failure and should be abandoned, As columnist Alfredo Relano wrote.
"Mourinho's latest pirouette was his worst," Marca's Santiago Segurola added.
"Spanish football has seen many teams with far fewer resources who have made life more difficult for Barca. And with greater nobility."
Real's next game is a La Liga clash at home to Athletic Bilbao on Sunday.
