But Mr Hamilton said visit by Mr Trump to Wales should be "encouraged".
The assembly said Ms Jones was duty bound to uphold the assembly's values.
In a letter to Ms Jones, Mr Hamilton said he was concerned by Ms Jones' comment on Twitter.
"He is now the elected President of the United States of America, and is actively seeking a trade deal with the UK," he said.
"The USA is Wales's most important global ally and trading partner, with 22% of Welsh exports going to the United States.
"It would be highly irresponsible to put Welsh jobs at risk by banning President Trump from visiting the Senedd.
"In fact we should actively encourage the UK Government to add Wales to the itinerary of his upcoming state visit."
"We are the National Assembly for Wales, not a student debating society," he said, adding that the rules of the assembly make clear the presiding officer must demonstrate impartiality.
"Mr Bercow has been accused of breaching the impartiality rule," he said.
"I hope you will agree that such controversy surrounding the office of Llywydd [the presiding officer] would be undesirable; not least because it could easily lead to calls for you to consider your position," he added.
'Respect'
A spokesman for the assembly said: "The Llywydd has demonstrated her respect and adherence to the principle of political impartiality consistently in her dealings with all Assembly Members."He added that Ms Jones was "duty bound to uphold the values which underpin the assembly's constitution and the Assembly Commission's work, such as our support for this week's LGBT History Month celebrations".
He said there had been no official request or invitation for the assembly to take part in the state visit of Mr Trump.