The leader of the Commons, Andrea Leadsom, has apologised to the Liberal Democrat deputy leader, Jo Swinson, for breaking a “pairing” pact during Swinson’s maternity leave but resisted calls for a proxy voting system to be introduced immediately.

It was “not good enough” that Swinson’s designated Conservative pair voted in critical Brexit votes on Tuesday, one of which the prime minister had won by just six votes, Leadsom told the Commons.

Under the formal arrangement, the Tory chairman, Brandon Lewis, had agreed not to vote to cancel out the fact that Swinson was at home with her three-week-old baby. But he then took part in two crunch votes on the government’s Brexit trade plans.

Leadsom, responding to an urgent question from the Lib Dem chief whip, Alistair Carmichael, said: “I’m very sorry that it happened. I’m assured by the chief whip that the breaking of the pair yesterday was done entirely in error and will not be repeated.”

She said that since Lewis’s vote did not affect the result there was no need for further action, but it was “vital” that maternity leave should be respected.

Carmichael noted that Lewis had voted at 6.15pm and 6.30pm, having failed to do so at 6pm and earlier in the day. “It does remain less than clear to me how this came to pass,” he said. The Lib Dem chief whip was making the point that the votes Lewis turned out for were in fact the most important that day.

More generally, he argued, pairing was “symptomatic of a wider problem”. He said: “It is using a 19th-century practice for providing cover under 21st century employment law, and that is no longer good enough.”

Valerie Vaz, Labour’s shadow leader of the Commons, said that rather than debating proxy voting in the autumn, as suggested by Leadsom, this could happen before recess next week.

A proxy system could even be introduced immediately, she suggested, with the votes of MPs absent on parental leave marked with a “P”, or even a baby emoji. “It is time that we ensured this is a modern workplace with modern employment practices,” she said.

Leadsom said an immediate change could bring “unintended consequences”, while debating the subject next week was not sufficient notice for MPs.

Julian Smith, the chief whip, said Lewis had been asked to vote “in error”. The Tory chairman said it had been an “honest mistake” by his party’s whips. Both men apologised.

Swinson, who later told party members in an email that she was “lost for words” and then “angry”, said it was time to give MPs a vote on allowing new parents to nominate a proxy vote on their behalf. She tweeted: “Enough of this. Instead of closing parliament early, let the Commons decide on proxy voting next week – with binding motions – and put an end to this charade.”

Labour’s Harriet Harman is among senior politicians who have been pushing for a formal parental leave system for MPs. The Commons backed the principle in March.

The cross-party procedure committee has already set out how such a system could be set up, including proxy voting for MPs who are absent after giving birth or through illness, but no action has yet been taken.

After the trade bill vote, a furious Swinson tweeted: “Just how low will your govt stoop Theresa May. When Andrea Leadsom delayed proxy voting motions, she *assured* those of us who were pregnant that we would be paired when necessary. Today your government broke that agreement. Desperate stuff.”

Lewis responsed: “I’m sorry Jo. I think it was an honest mistake made by the whips in fast-moving circumstances. I know how important the pair is to everyone, especially new parents, and I apologise.”

MPs across the Commons were angered by the breach. Dr Sarah Wollaston, a leading Tory remainer who voted against the government, asked who was taking responsibility for failing to honour the pairing agreement for Jo Swinson. “More than just an extension of the other heavy-handed tactics on display, it disrespects women and why maternity leave matters.”