Performance director Wotte made his comments on Twitter after 29-year-old midfielder Commons announced he had quit Scotland on the same day national team boss Gordon Strachan left him out of his squad for next month's World Cup qualifier in Croatia.
Wotte said: "Retiring at 29 after only 10 caps honestly still puzzles me. Where does pride, desire and ambition come into this picture? Agree?"
Scotland midfielder Brown, however, supported his Parkhead colleague, who won two more caps than Wotte gave him credit for.
Brown told a number of newspapers: "There is nothing wrong with Kris' desire. He has come to a club such as Celtic and shown what he can do.
"Kris trains hard, he's always in the gym.
"He is one of the first ones into training in the morning and one of the last ones to leave, so there is nothing wrong with his desire whatsoever."