The Leveson Inquiry has been warned that actor and activist Hugh Grant has been “punished” for his decision to speak out about press intrusion on Monday.

Associated Newspapers – publisher of the Daily Mail and the Mail on Sunday- hit out at Grant’s allegation that his phone had been hacked by the Mail on Sunday by claiming that he had lied under oath.

“Mr Grant’s allegations are mendacious smears driven by his hatred of the media,” the publisher said in a statement that also said that it “utterly refutes” Grant’s claims.

Barristers representing the Metropolitan Police and numerous hacking victims have said that the actor’s treatment after giving evidence would probably be “intimidating” to other witnesses.

The Daily Mail also “unequivocally denies” the paying of a hospital source to secure information about the birth of his first child.

Robert Sherborne, the legal counsel representing the hacking victims said the newspaper was using its power to bully the star.

“What was filed in the pages of the Daily Mail and the website was not a denial but a personal attack on Mr Grant as a witness…what they suggested is that he was deliberately lying.”

“There is a difference between a right of reply and a right of attack – if those you have been brave enough to come and give evidence receive this kind of treatment then witnesses will be unwilling to be that brave any longer,” Sherborne added.

Metropolitan Police QC Neil Garman was also concerned that Grant’s treatment could affect other witnesses

“Witnesses will be very cautious, we fear, if the likelihood is that they will face that kind of treatment the day after.

Other high profile witnesses who are expected to give evidence are Steve Coogan, Sienna Miller [one of the first people to sue News Corporation] and JK Rowling.

Jonathan Caplan QC, representing Associated Newspapers, said that the newspapers had made the statement “under pressure” in response to serious allegations of serious criminal misconduct, which it denied.

When asked about the attack on Grant, Caplan told Lord Justice Leveson:”I accept everything you say.”

Grant told the hearing he had tried to keep his ex-girlfriend Tinglan Hong’s pregnancy secret after the News of the World first speculated that she may be carrying his baby in April. Grant said: “my overwhelming motive throughout this whole episode was to protect the mother of my child”.

Grant said that Hong had been followed by paparazzi but the papers “seem not to have anything to print that could link her to me until I visited the hospital after the birth when again there seems to have been a leak from the hospital”

Grant said that he received a phone call the next day from the Daily Mail, and then the Daily Star. He said that he refused to comment, believing that to do so would give the paper confidence to publish.

He said that the Mail had adopted a “fishing technique and they didn’t want to print the story based solely on the hospital source because that might have been unethical or possibly illegal so they needed a comment from my side and that is why I said nothing”.

The Mail did not publish the item. The news was then published by US Weekly last month. The actor said the baby was the product of a “fleeting affair” and that Hong was “besieged” by photographers after the news was published and the actor was forced to take out an injunction to force them to go away.

When questioned by Robert Jay, QC to the Leveson inquiry, Grant said the only people who knew about he child were a female cousin, and Hong’s Chinese parents, “who spoke no English”.

The Mail issued a statement saying it “unequivocally denies Hugh Grant’s allegation that it secured information about the birth of his child from a source at the hospital” which instead came from “a source in his showbusiness circle more than two weeks after the birth”.

Grant said that he was sure the Mail had hacked his phone and referred to a report in the Mail on Sunday in February 2007, which said his relationship with his then girlfriend, Jemima Khan, was “on the rocks” because of “persistent late-night phone calls with a plummy-voiced executive from Warner Brothers” – a story he said was “completely untrue”. Jemima Khan also denied the story via her Twitter account.

Lord Justice Leveson also heard evidence from the parents of Milly Dowler, the murdered schoolgirl whose phone was appallingly hacked by the News of the World.

Bob and Sally Dowler spoke publicly for the first time about the moment they believed their daughter was picking up her voicemail messages, giving false hope that she was still alive.

Sally Dowler’s told the inquiry that after a period in which every time she rang her missing daughter’s mobile phone, it said the mailbox was full. She said: “It clicked through on to her voicemail so I heard her voice and [said]: ‘She’s picked up her voicemail Bob, she’s alive’.”

The couple also told the court that a private walk they took seven weeks after their daughter’s disappearance was pictured prominently in the News of the World. They claimed photographers were tipped off about the walk after the paper hacked their mobile phones.

Bob Dowler said: “The thing to remember is the walk was nothing to do with Milly’s phone.” His wife added: “That was our own home phone or own mobile phones.”

News International has paid £2m in compensation to the Dowler family, and the company said it had nothing further to add following their testimony.