Dyke's comments on his show on Wednesday also included the assertion that "yummy mummies" should not breastfeed in public, which angered some listeners and prompted complaints to the regulator Ofcom.
On Thursday, Dyke apologised on air, saying: "The comments I made during the broadcast were unacceptable and I would like to apologise for any offence caused."
He has now been suspended from the BBC, with a spokesperson for the corporation issuing the statement: "Following unacceptable comments made on air, Alex Dyke has been suspended pending an investigation." The spokesperson declined to elaborate further.
In addition, communication regulator Ofcom said it was looking into the issue after receiving 32 complaints, which will be assessed within the next week along with the recording of the episode to decide whether to proceed with an investigation.
The comments spurred an online petition to have him fired that has gained more than 10,000 supporters thus far. Maisie Jameson, creator of the plea, wrote: "There is already a stigma around breastfeeding, and we don’t need ill-educated misogynists like Alex Dyke to make it even more apparent. Breastfeeding is completely natural, and people must remember that breasts are for feeding – not for men."
Others have criticised Dyke and the BBC on social media over the episode, which has since been removed from BBC iPlayer.
Breastfeeding in public has frequently proved a difficult topic for public figures, with UKIP leader Nigel Farage criticised over comments on the matter last year, while earlier in 2015 Facebook updated its policies on a number of areas including users posting pictures of breastfeeding.