They are unhappy that he failed to pay his respects at the grave of former President Ziaur Rahman on the anniversary of his assassination last month.
Ziaur Rahman, who is revered as the founder of the BNP and was Prime Minister Khaleda Zia's husband, was killed in an abortive army coup in 1981.
"We do not understand what had prevented Chowdhury from visiting the grave," said BNP chief whip KD Hossain.
Dr Chowdhury confirmed his resignation, which had been expected for 24 hours, in a letter to the Speaker of parliament on Friday.
"I have resigned according to the ruling party's decision, to uphold the democratic process," Dr Chowdhury told the BBC.
"There was no mention of my not having complied with the protocol in the letter seeking my resignation.
"Nor did I receive any resolution from the party on this. I only read newspaper reports on the party's anger," Dr Chowdhury added.
Presidential neutrality
He said that since the presidency demanded neutrality, he could not attend party-specific programmes while in office.
"That in no way means that I do not respect the late Ziaur Rahman or the Bangladesh Nationalist Party, for which I have worked over the last 24 years," Dr Chowdhury said.
Earlier, his spokesman said he had thanked palace staff and received his last military honour guard before signing the resignation letter.
He and his family were then driven from the palace to his private residence in a car that did not carry the presidential standard.
Dr Chowdhury, 71, is a former foreign minister and founding secretary-general of the Bangladesh Nationalist Party, who was elected president by the ruling coalition seven months ago.
'No disrespect'
BNP leaders were angered by Dr Chowdhury's absence at Ziaur Rahman's grave, and by his failure in a message on the anniversary to say that the BNP's founder had declared national independence.
The leader of the opposition Awami League, Sheikh Hasina, declined comment, but her chief whip has described the BNP's move as a "big threat" to democracy.
The Bangladeshi press has been almost uniformly critical of the ruling party's demand that the president step down.
The president's son insists, however, that he did nothing wrong.
"My father did not disrespect Ziaur Rahman and he is stepping down due to his respect for democracy," Mahi Chowdhury, himself a BNP MP, said.
"The party was more important to him."
Historical dispute
The current speaker of Bangladesh's parliament, Jamiruddin Sircar, will serve as acting president until the parliament elects Dr Chowdhury's successor within 90 days.
Mr Sircar rejected criticism that the president's resignation was a setback for constitutional democracy.
"The process is very much in tune with parliamentary democracy where the presidency is largely an ornamental post," he told the BBC.
"It shows democratic government is in place."
The identity of the person who declared independence from Pakistan in 1971 is a source of impassioned controversy in Bangladesh.
The BNP says Ziaur Rahman, then an army major, declared independence in March 1971 - while the Awami League says its then leader, Sheikh Mujibur Rahman, was responsible.