The lord chancellor has said she takes her oath to defend the judiciary seriously but will not criticise the press for the reporting of the high court Brexit case, in which three judges were called “enemies of the people”.

In a letter to the Times, Liz Truss said the independent judiciary was robust enough to withstand attack by the Daily Telegraph and Daily Mail.

Last week, one of her predecessors, Lord Falconer, called for her to be sacked as justice secretary. He wrote to the Times that her failure to speak out “signals to the judges that they have lost their constitutional protector”.

“Until she is replaced by someone willing and brave enough to do the job, the judges should rightly fear for their independence,” Falconer said.

Truss has faced mounting criticism in legal circles, including from her own party, for her perceived unwillingness to oppose attacks on the judges who decided that the two-year formal process for leaving the EU could not be triggered by the government alone and required a vote from parliament.

In her reply to Falconer on Thursday, Truss denied she had failed to defend the judges. “An independent judiciary is the cornerstone of the rule of law, vital to our constitution and freedoms,” Truss wrote.

“It is my duty as lord chancellor to defend that independence. I swore to do so under my oath of office. I take that very seriously and I will always do so.

“Lord Falconer of Thoroton QC says that ‘judges should rightly fear for their independence’ due to my not condemning the reaction of some newspapers to the Brexit ruling. I think it unlikely the high court is imperilled by the opinions of any newspaper.”

Truss said the freedom of the press was also an important principle that was at stake in the debate. “I believe in a free press, where newspapers are free to publish, within the law, their views,” she said. “It is not the job of the government or lord chancellor to police headlines, and it would be a dark day for democracy if that changed.”

Truss called the three judges, including the lord chief justice, who ruled on the case “people of integrity and impartiality, as are the 11 supreme court justices who will hear the appeal”.

The trio exercised that independence and made a ruling without fear or favour, she said. “The right of the judges to make that judgment was never in question, and I defended their independence after that decision.”

Tory MPs and peers have also made their disquiet known to Truss, meeting her on Monday night to voice concern about the slowness of her response. Those at the private meeting said there was astonishment when Truss suggested the job of defending the judiciary should fall to the lord chief justice in the first instance, given that he had sat on the case and was one of those under attack for the judgment.

In the House of Lords on Tuesday, the Conservative former justice minister Edward Faulks, a past critic of Truss, called the past week’s events “a disgraceful attack on the judiciary”.

The Conservative chair of the Commons justice select committee, Bob Neill, issued a statement too, saying that a right to freedom of the press “should not be couched in terms of abuse of individuals who, by virtue of the oath they have taken and the role they discharge, cannot defend themselves publicly”.