But Puigdemont called for an global arbiter to mediate the crisis, ideally the European Union.
Pique also criticized Spanish Prime Minister Mariano Rajoy for the way the government and police dealt with Sunday's plans for a referendum, which was ruled illegal by the highest rung of Spain's judiciary, the Constitutional Court.
In a rare nationwide address King Felipe launched a blistering attack Catalonia's pro-independence regional government, accusing the secessionists of "breaking democratic principles" and trying to split up Spain.
Catalan officials said 90 per cent of the 2.26 million people who took part in the vote backed independence.
Hundreds of thousands of people across Catalonia have been protesting over Spanish police violence during the vote, during which almost 900 people were hurt, including 33 police officers.
Catalonia, a region of 7.5 million people in north-eastern Spain, has its own language and culture and Barcelona is its capital.
Puigdemont said there was now no contact between the government in Madrid and his devolved administration.More news: Trump Administration Poised to Roll Back Federal Birth Control Coverage Mandate
The king's speech was a response to the thousands of Catalan residents that hit the streets to protest the violent crackdown inflicted over the weekend by Spanish authorities attempting to block the vote, according to the outlet.
"I think I can stay in the national team because I truly believe there's many people in Spain who disapprove of what happened today in Catalonia and who believe in democracy", he said.
"Despite all of the challenges and the horrendous scenes we saw in Catalonia yesterday, the overwhelming majority of those who voted, voted for independence".
Catalan President Carles Puigdemont has said he does not want a "traumatic break".
The Foreign Office said the referendum was "a matter for the Spanish government and people".
If the Catalan government unilaterally declared independence from Spain, Rajoy could suspend the President and take over the running of the province. During the day Spanish riot police smashed their way into polling stations to try to halt a disputed independence referendum and fired rubber bullets at protesters.