"Anyone who is proven to have participated in the criminal acts will be punished with the full force of the law."
He wrote that he defended people's right to protest, "but neither can we allow people to vandalise public or private property".
The president said he had been meeting various groups to present changes to the controversial budget.
Discontent had been building over the 2021 budget on social media and clashes erupted during demonstrations on Friday. Guatemalans were angered because lawmakers approved $65,000 to pay for meals for themselves, but cut funding for coronavirus patients and human rights agencies.
Vice President Guillermo Castillo has offered to resign, telling Giammattei that both men should resign their positions "for the good of the country". He also suggested vetoing the approved budget, firing government officials and attempting more outreach to various sectors around the country.
Giammattei had not responded publicly to that proposal and Castillo did not share the president's reaction to his proposal. Castillo said he would not resign alone.
The spending plan was negotiated in secret and approved by the congress before dawn on Wednesday. It also passed while the country was distracted by the fallout of hurricanes Eta and Iota, which brought torrential rain to much of Central America.
The Roman Catholic Church leadership in Guatemala also called on Giammattei to veto the budget on Friday.
"It was a devious blow to the people because Guatemala was between natural disasters, there are signs of government corruption, clientelism in the humanitarian aid," said Jordan Rodas, the country's human rights prosecutor.
He said the budget appeared to favour ministries that had historically been hotspots of corruption.
