“Your definition of antisemitic language is different from ours,” he said in the letter, which the Board published on Wednesday.
He accused Mrs van der Zyl of misrepresenting him, saying it would have been "correct from you to state that my opinion was a different one”.
He also accused Mrs van der Zyl of not having the “slightest respect" for Hungary.
He added: “I harshly reject again your allegations of antisemitism against us, and especially against PM Orbán. It is a shame for you and a shame for the whole cause: a simple lie.”
Mr Szalay-Bobrovniczky was particularly angered by Mrs van der Zyl's comments about Mr Orbán's rhetoric on Mr Soros, the Hungarian-born billionaire philanthropist whose support for refugee rights made him a target of the prime minister.
The PM once said of Mr Soros in a speech: "We are fighting an enemy that is different from us. Not open, but hiding; not straightforward but crafty; not honest but base; not national but international; does not believe in working but speculates with money; does not have its own homeland but feels it owns the whole world."
Mr Szalay-Bobrovniczky accused Mr Soros of organising “illegal migration to Europe” and “undermining European security and the security of Jewish life on our continent”.
He added: “That is our problem. I am extremely sorry for the Jews that you pretend to represent.”
He told Mrs van der Zyl to “mind your own business”. In the press release after the meeting, Mrs van der Zyl had described the conversation as “respectful and honest.”
She said it would be “a clear test on the Hungarian Government’s good-will on the history of the Holocaust... whether the new museum, will command the support of the mainstream Hungarian Jewish community.”
She also had referred to the Hungarian Government's “moves to adopt the IHRA definition of antisemitism.”
Mr Szalay-Bobrovniczky said this was not what he said, adding: “The IHRA working definition is being worked at and will probably be soon passed by the Government: these were my words, not what you stated."
He accused her of being “everything but respectful and honest”, claiming she was guilty of "relativisation of the obviously antisemitic tendencies in the Labour Party” and said she was “politically motivated”.
He claimed he had met conservative peers Lord Pickles and Lord Ahmad and said neither thought "we would speak or act in an antisemitic way".
"That is decisive for us," he added. "Your politically motivated, your own Labour political side whitewashing opinion remains a single one in the UK."
Lord Pickles confirmed that he met Mr Szalay-Bobrovniczky before Christmas, as the UK Special Envoy for Post-Holocaust Issues.
They discussed the controversial House of Fates museum in Budapest, which critics have argued minimises the complicity of Hungarians in the Holocaust, and the Hungarian government’s adoption of the IHRA working definition of antisemitism.
Lord Pickles said of Mr Szalay-Bobrovniczky's claim: “Things sometimes get lost in translation”.
He told the JC: “It was a business-like meeting. It lasted half an hour and we had those issues to go over, we didn’t talk generally about Orbán.
“Let’s put it this way: things sometimes get lost in translation, but it was a business-like, diplomatic meeting. We had things of substance to talk about.
“We did not get into generalities. Perhaps what he meant was that we got on very well together, and I’m hoping to go to Hungary fairly soon, again to talk about the House of Fates.”
Mrs van der Zyl wrote to Mr Szalay-Bobrovniczkythat she was "astonished and disappointed" by his words.
"With all the positive things to say, we do not want to have to criticise Hungary for the negatives we see, and we only ask that you take heed of our concerns and take action to address them," she said.
She doubled down on her claim that Mr Orbán's rhetoric "strays in to old antisemitic tropes and we urge the Prime Minister and the Government to step back from, and apologise for, this sort of language".
"I urge the Hungarian Government to take heed of our clear but constructive comments, to ensure that the international reputation of Hungary can once again be of a country that treats Hungarian Jews and other minorities with the equal dignity and respect that we must all want for everyone," she said.
