Former world time trial champion Pooley claimed British Cycling - investigating claims of sexism and discrimination in the sport - should also look into why no women's team was formed by Team Sky.
Kennaugh tweeted Pooley should "get over it" before deleting his message.
He later tweeted an apology, saying he came across like a "total idiot".
Pooley, who twice finished second in the Giro d'Italia, believes Team Sky boss Sir Dave Brailsford, the former performance director of British Cycling, should have offered more support for a professional women's road race team.
The 33-year-old Olympic silver medallist told the Guardian: "I wish more questions had been asked of him before he was awarded his knighthood and moved to Sky.
"It was when he was running British Cycling that there was no women's Team Sky."
Kennaugh had earlier supported Shane Sutton, who resigned as British Cycling technical director amid allegations of sexism and discrimination towards elite cyclists.
Kennaugh, who won team pursuit gold at London 2012, told BBC Radio York: "Shane's a great guy with a massive heart.
"It is a sport and if you are not meeting the standards, which are very high at British Cycling, then unfortunately there's no place for you anymore.
"It is ruthless at this level. I think a lot of it has been blown out of proportion."
Double British road race champion Peter Kennaugh has apologised to Emma Pooley over comments he made as part of the ongoing sexism row in British cycling.