Czarzasty said on Monday that he would not support a "reset" or changes to the constitution, insisting that decisions made by parliament must be respected.
The "reset" proposal would involve replacing all current Tribunal judges with a new bench elected by a cross-party supermajority.
The Sejm, lower house, has voted in a total of eight judges to the court in recent months.
Nawrocki has refused to receive oaths from five of them.
Czarzasty accused the president and the previous Law and Justice (PiS) government of contributing to the Tribunal's paralysis, arguing that all judges elected by the current parliament should be allowed to take up their duties.
The dispute escalated on Friday, when seven of the newly elected judges entered the court's headquarters in Warsaw, according to the spokesperson.
They announced that senior judge Anna Korwin-Piotrowska would temporarily lead the institution and suspend the duties of its president, Bogdan Święczkowski, and vice-president, Bartłomiej Sochański.
Święczkowski rejected the move as illegal, claiming that he remained the only authorised administrator of the building.
Nawrocki, who cut short a visit to Senegal to return to Warsaw, later accused the judges of forcibly taking over a constitutional state institution.
According to Prime Minister Donald Tusk, they were prevented from taking up their posts and attacked when they attempted to do so.
He described the incident as another threat to the rule of law in Poland.
Meanwhile, Zbigniew Bogucki, head of the President's Chancellery, accused Tusk of crossing "all boundaries" over the events at the court.
He called the presence of police at its headquarters and the actions taken there as "absolutely unprecedented."
Czarzasty said he would also use his constitutional powers to file a complaint with the Tribunal, adding that the authorities must take action to restore its proper functioning.
(ał)
Source: PAP