The incident occurred in the 62nd minute when Rooney broke through two tacklers, then stomped on the prone Carvalho's groin. England went on to lose the quarter-final in a penalty shootout.
"I remember the incident clearly and have seen it several times since on TV," the Manchester United striker said in a statement. "I am of the same opinion now as I was at the time, that what happened didn't warrant a red card. If anything I feel we should have had a free kick for the fouls committed on me during the same incident.
Portuguese winger Cristiano Ronaldo ran up to the referee to protest Rooney's actions, getting a shove from his United teammate in the process.
"When the referee produced the red card I was amazed, gobsmacked," Rooney said. "I bear no ill feeling to Cristiano (Ronaldo) but I am disappointed he chose to get involved. I suppose I do, though, have to remember on that particular occasion we were not teammates."
Ronaldo, who was caught winking to the Portugal bench after Rooney was ejected, denied a rift between the two.
British papers reported the Ronaldo and Rooney statements and text messages appear to be an attempt at a peace bid brokered by United manager Sir Alex Ferguson.
But The Sun reported that Rooney still is in no mood to make up, saying he had confided to teammates the only way he would like "to bury the hatchet is in Ronaldo's head."
Elizondo, speaking to The Times, said he sent off Rooney for lashing out with his boot and catching the defender in the groin, not for the 20-year-old's push on Ronaldo, accoridng to the AP.