The short answer is no, because International Men's Day is not an observed international day as declared by the United Nations General assembly (but International Women's Day is). International Men's Day was invented by a college professor at Missouri University with no international standing.
Google has to make some concessions about which doodles they make and which they don't, as each day is overloaded with holidays. November 19 is also Feat of Obadiah, Feast of Raphael Kalinowski, Feast of Severinus, Exuperius, and Felican, Day of Discovery (Puerto Rico), Day of Missile Forces and Artillery (Russia, Belarus), Flag Day (Brazil), Garifuna Settlement Day (Belize), Liberation Day (Mali), Martyrs' Day (Uttar Pradesh, India), The Sovereign Prince's Day (Monaco), Women's Entrepreneurship Day, and World Toilet Day.
If you don't like it, make a change.org petition or reach out to Google directly.