He absolutely can do that, but is there a majority to extend article 50 to enable parliament taking control of the process and agreeing on something? The "people's vote" crew can't even agree on wording, let alone what to do. I think Maybot worded stuff in a way by delaying the vote that it makes it hard. A rare bit of forethought.
The EUs intransigence on helping Maybot has really limited the options. A little assistance on the wording on the backstop would help her, but they pushed her head further into the dirt. They are almost forcing a No Deal on us
There is without a doubt a majority to stop a No deal Brexit. If the Speaker was to take action then I suspect a majority across the house would emerge to extend A50 for a more ordered approach rather than the farce that is playing out due to the intransigence of May and Corbyn. Remember, Corbyn is desperately trying not to bring a no confidence vote against the Govt as when he loses and fails to get a GE then automatically Labour's policy switches to pushing for a 2nd referendum. Bercow taking action would sideline the bottleneck of May's and Corbyn's personal pride.
As for the EU's intransigence on the backstop, they have been consistent which has funnily enough followed the principles of the Good Friday Agreement. All the backstop issues now culminate from how May has acted before and since triggering A50, she has brought it all on herself. May agreed the legally binding wording weeks ago so she now can't complain about it after months of negotiations.
Why is it "intransigence" by the EU? They negotiated in good faith with May and May knew the opposition in Parliament to such a position (especially as she is reliant on the DUP* and they haven't been shy in telling her) so she should have factored it in. The EU offered to do a supplementary clarification but that would never satisfy the ERG or the DUP as rightly it would never supercede the legally binding agreement. If May hadn't been so obsessed with her own party and dealt with the House as a whole trying to build a consensus across the house early in the process, rather than the ridiculous GE and pontificating since triggering A50, then we may have had an orderly exit.
Also, the EU is rightly looking to protect its member, which Ireland is.
*It is ridiculous being reliant on a party who basically want her to square a circle.