"I cannot give the mandate to somebody who threatens the sovereignty of Macedonia," said President Ivanov.
Social Democrat leader Zoran Zaev formed the coalition after agreeing to support a bill making Albanian the country's second official language.
Thousands of Macedonians have been protesting against the bill.
An official from the Albanian Democratic Union of Integration (DUI), which had hoped to join the coalition, said the situation was "chaos".
The former Yugoslav republic has been in political crisis for two years following a phone-tapping scandal.
December's general elections were brought forward two years as part of a Western-brokered agreement, but failed to produce an outright winner.
The results between the governing VMRO-DPMNE party and the Social Democrats (SDSM) were so close that both parties initially claimed victory.
Forming a coalition with minority parties has been their only way forward.
The VMRO-DPMNE, which won two more seats than the opposition, has already tried but failed to form an alliance.
Macedonia came close to civil war in 2001 after an Albanian uprising.
Ethnic Albanians make up around a quarter of the country's population.
Macedonian President Gjorge Ivanov has refused to grant a coalition of Social Democrats and ethnic Albanian parties a mandate to form a government.