Following Saturday's 3-1 win over the St. Louis Cardinals in Game 2, Doolittle addressed the situation involving his teammate.
"If your reaction to someone having a baby is anything other than, 'Congratulations, I hope everybody's healthy!' you're an assh--e," Doolittle told reporters.
The Nationals announced Hudson would miss the NLCS opener Friday because his wife gave birth to the couple's third child.
Former Miami Marlins president David Samson was among the people critical of Hudson for missing a baseball game to be with his wife.
"Unreal that Daniel Hudson is on paternity list and missing game 1 of #NLCS," Samson tweeted. "Only excuse would be a problem with the birth or health of baby or mother. If all is well, he needs to get to St. Louis. Inexcusable. Will it matter?"
The Nationals won 2-0 in Game 1 thanks to Anibal Sanchez's no-hit bid that got broken up with two outs in the eighth inning. Doolittle followed him out of the bullpen by getting the final four outs.
Hudson returned to the Nationals in St. Louis for Game 2. He recorded the final two outs to give Washington a 2-0 series lead heading back to Nationals Park.