"We're not going to give the nod to everyone," Erdogan said on Sunday. "We also have certain rights."
Kosaner will take command after General Ilker Basbug retires later in August.
Last month, a Turkish court indicted 196 people, including four retired military commanders, over an alleged plot to topple the government. The arrest warrant was later overturned.
Hundreds of other people have been arrested in connection with various anti-government plots over the last two years.
Turkey's military has unseated four governments since 1960, under the auspices of protecting the country's secular constitution. It has often clashed with Erdogan and his Justice and Development Party.