The large brawl involved nationals from Syria and several African countries at an overcrowded migrant reception camp outside of Cyprus’ capital Nicosia.
It comes after Cyprus began its COVID vaccine programme on Sunday.
The measures were set to span the Christmas holidays and will stay in force until December 31.
The Turkish president's comments came during a controversial visit to the north of the island, occupied by Turkey since 1974 and not recognised internationally.
European diplomats say Cypriot officials are threatening to veto sanctions against Minsk if the EU does not consider action over a separate dispute in the Mediterranean Sea.
US Secretary of State Mike Pompeo has urged Turkey to engage in dialogue and take no more actions to increase tensions with Greece in the Eastern Mediterranean.
As tensions continue to rise in the eastern Mediterranean, with Turkey, Greece and Cyprus at odds over territorial waters, the Turkish-Cypriot leader told Euronews EU countries should be working to de-escalate the situation.
It comes after a number of migrants who recently crossed from the breakaway north to the internationally recognised south of Cyprus tested positive to COVID-19.
The government of Cyprus is encouraging the voluntary use of a locally developed app designed to locate people who may have come into contact with someone infected with coronavirus.
British woman given four month suspended sentence over 'false' rape claim
A series of blasts hit an ammunition depot in northern Cyprus after a fire broke out in a military zone.
A 96-year-old World War Two veteran celebrated his birthday on Saturday (August 31) by breaking his own record as the world's oldest active scuba diver for the third year running.
A Turkish ship planning to drill for oil and gas close to Cyprus dropped anchor off the island on Monday, sparking tensions with Nicosia and the European Union.