Labour’s deputy leader, Tom Watson, has stepped up his criticism of Jeremy Corbyn over antisemitism in the party by backing calls for automatic exclusion of members accused of anti-Jewish racism.

He also warned the Labour leader he was determined to call out the problem every day until the introduction of meaningful reforms also including an independent system for handling complaints of antisemitism.

John McDonnell, the shadow chancellor, has called on the BBC to carry out an investigation into complaints made by Labour about the Panorama report into antisemitism in the party.

“There have been complaints put in to the BBC now. I think the BBC should investigate those and then we can come to a conclusion,” he said. “What we have got is ex-staff making accusations against existing staff and those existing staff have challenged those complaints, so it does need an objective look at.”

Speaking to BBC Radio 4’s Today programme on Thursday, Watson urged Corbyn to immediately introduce changes including automatic exclusions and said the Labour leader was the only person who could tackle the problem.

“Jeremy has the votes on our NEC [national executive committee]; if he willed it, it would happen … Not only do I think he can fix it, I think he’s the only one that can fix it, he said.

“I am not going to turn a blind eye to anti-Jewish racism, I’m going to call it out day in day out until action is taken. And that might cause very great difficulty for my colleagues in the shadow cabinet, who are also collectively responsible for this. But until we’ve dealt with it, until we’ve actually changed our rules, until we’ve actually attacked the culture at its root cause then I’m not going to resile.”

Watson said he deplored the way in which the leadership had responded to claims from whistleblowers that attempts to deal with the problem had been undermined by senior figures in the party.

And he said he and the shadow cabinet had been denied access to information on the scale of the problem and the party’s official response to an investigation by the equalities watchdog into how Labour handles allegations of antisemitism.

Watson pointedly praised former party officials who broke non-disclosure agreements on the Panorama programme about interference by the leader’s office in investigations into antisemitism claims.

Labour complained to the BBC about the programme and dismissed the whistleblowers as “disaffected employees”. But Watson said: “I deplore the statement that was made about those people last night; you couldn’t fail to be saddened and moved by the testimony, particularly of the young members who’d had racist abuse in party meetings.”

He said the whistleblowers “showed great courage … and to dismiss their testimony as in some way flawed, I think was wrong. We’ve got to agree that the Labour party will never again use gagging orders to try and stop people expressing their deeply held concerns about racism in our own ranks.”

He said the whistleblowers’ claims showed the need for “a full independent system of investigating cases of anti-Jewish racism that involves representatives from the Jewish community and Britain of standing”.

And he backed calls by Keir Starmer, the shadow Brexit secretary, and the former prime minister Gordon Brown for a party rule change to automatically exclude members “who have a prima facie case to answer of using antisemitic behaviours and language”.

Watson added: “That effectively means people who are facing an investigation of antisemitism can’t go to meetings any more where Jewish members feel threatened and insecure and intimidated.”

The deputy leader said he could not assess whether Panorama exaggerated the scale of antisemitism in the party because he was unable to find out how many complaints had been made.

Watson claimed that since Corbyn was elected leader, there has been a “growing belief that there is a sickness in our party, that this kind of abuse has been in some way allowed, that there’s almost a permissive culture that people can use anti-Jewish, racist language … that we failed to address adequately.”

He said he was “very concerned” he and other members of the Labour NEC had not been given access to the party’s response to an investigation by the Equality and Human Rights Commission into how it handles allegations of antisemitism.

“I’ve been told by a general secretary that I’m not legally responsible for dealing with outside regulatory bodies,” Watson said.

He also said he was “deeply concerned” by whistleblower allegations that emails from Labour’s general secretary, Jennie Formby, about investigations into antisemitism claims were deleted, after he had been assured by Formby they had been preserved.