Ms Butler had said: “Poor people in our country have paid with their lives because the Prime Minister has spent the last 18 months misleading this House and the country over and over again. It’s dangerous to lie in a pandemic."
“I don’t buy into this argument that character just doesn’t matter," he said, adding that in the end "integrity, honesty and accountability" will out.
Last month it was hinted that the prime minister's former aide could join Labour if the right opportunity arose.
Owen Jones tweeted: "Keir Starmer promised no more navel gazing, but instead of fighting the Tories, Labour is now beginning a Stalinist style purge..."
“Far from giving racism the red card, the prime minister gave it the green light," Starmer said.
Theresa May also gave an impassioned speech announcing she will be voting against the cuts to the aid budget - the first time an ex-PM has voted against a Conservative three-line whip in 25 years.
The Labour leader said he was shocked by reports that ministers had decided to “tune” the app.
Kim Leadbeater, sister of murdered former constituency MP Jo Cox, squeezed home in the contest by just 323 votes.
Brendan Cox, husband of murdered Jo Cox, said the people of Batley and Spen had "voted for decency and positivity once again".