He added that Golden Dawn aligned itself with statements made by Donald J. Trump, who had called for Muslims to be barred from the United States. “It would be a very positive development if he were to be elected president,” Mr. Kasidiaris said.

The clashes in Idomeni unfolded when a group of migrants acting as emissaries for those in the Idomeni camp approached the barbed-wire-topped border fence and asked the Macedonian police to open it to let them continue north. When the request was declined, the men apparently threw rocks at the police, who responded by firing tear gas at the growing crowd.

Doctors Without Borders and several other humanitarian groups in Idomeni said in statements that they were treating numerous people for injuries, including men who had been knocked unconscious as well as children, after tear gas blew into hundreds of flimsy pup tents where the migrants have been living in the mud over the last several weeks.

Under the deal that took effect last month, all migrants who arrived in Greece on or after March 20 are to be deported to Turkey. Migrants who arrived in Greece before March 20 are to have their applications processed in Greece and will not be deported to Turkey if the applications are approved.

Afghans and other migrants who were in Greece before March 20 but are unlikely to qualify for asylum in Europe can remain in Greece until their fate is decided, but if their applications are denied they will not be able to remain legally in Europe.

The Greek authorities have said they want to clear out the Idomeni camp soon and move people to nearby military-style encampments, which they say offer better shelter than the flimsy tents that now carpet the fields abutting the Macedonian border. Migrants, however, have been reluctant to leave, fearing that once they go, the authorities may try to shut them in the camps.

Officials also appealed over the weekend to the thousands of migrants camped in Piraeus to leave voluntarily or risk being relocated forcibly within the next two weeks. Violent brawls have broken out in the port and in migrant camps elsewhere in Greece recently, adding a sense of urgency to the situation.