The centre-back, 28, was jeered by a section of Spain fans during Saturday's Euro 2016 qualifier win over Slovakia, and was subjected to similar during the previous internationals break.
"The last thing I would do is leave the national team," he said.
"I won't change regarding club rivalry. I always wish Real Madrid to do badly."
It was reported that England's friendly in Spain on 13 November was moved from Real's Bernabeu Stadium to Alicante because of concerns over the possible negative reaction to Pique's involvement.
In June, the former Manchester United defender sarcastically thanked Colombian rapper Kevin Roldan for playing his part in Barca's success last season.
The performer posted photos on social media of Real team-mates celebrating Cristiano Ronaldo's 30th birthday in February just hours after a 4-0 defeat by Atletico Madrid. The players were consequently criticised.
Barcelona went on to win the league, Copa del Rey and Champions League.
"I don't regret anything that I have said and I would repeat it a thousand times and more," added Pique.
"I am like that, I want Madrid to always lose. That is the sporting rivalry that has always existed in Spain."

Mr Maduro issued a decree to convene a constituent assembly amid continuing anti-government protests.
The president says a new constitution will bring peace to Venezuela, but the opposition says it is a ruse to delay holding elections.
More than 50 people have been killed in protest-related violence since 1 April.
Mr Maduro signed the decree at a rally attended by thousands of his supporters.
He said the assembly would be made up of 540 members, some of whom would be elected at local level while the rest would be chosen from various groups such as students, workers, farmers, pensioners, and indigenous people.
The National Electoral Council later said that elections for the constituent assembly would be held at the end of July.
The move was immediately denounced by the opposition-controlled National Assembly.
National Assembly leader Julio Borges called it "nothing more than an evil announcement meant to divide, distract, and confuse Venezuelans further".
He said there would be more "street action" and called for a march opposing the constituent assembly to be held on Wednesday.
Fifty-five people have been killed in protests since the beginning of April.
The unrest was sparked by a Supreme Court ruling on 29 March stripping the National Assembly of its powers and transferring those powers to the court.
The Supreme Court suspended the most controversial paragraphs three days later, but the ruling united the hitherto divided opposition and spurred them into action.
The opposition says the Supreme Court ruling and Mr Maduro's call for a constituent assembly are attempts by the president to cling on to power beyond the end of his term in 2019.
They say that he has become increasingly authoritarian since he was elected in 2013, blocking their attempts to hold a recall referendum and delaying elections for governors.
On Tuesday, National Electoral Council head Tibisay Lucena also announced that the gubernatorial elections would be held on 10 December, a year after they were originally due.
Some Venezuelans reacted on social media saying that they had lost all trust in the electoral body and demanded fresh general elections rather than regional polls.
Barcelona defender Gerard Pique will stick by his anti-Real Madrid stance and continue playing for Spain despite being booed by supporters.
[NEXT_CONCEPT]
Venezuela's President Nicolás Maduro has pressed ahead with plans to create a citizens' assembly to draw up a new constitution.