Assange was arrested on Thursday at Ecuador's London embassy on allegations of skipping bail, and on a US extradition warrant related to a huge leak of official documents.
A letter written to Ms Abbott, signed by more than 70 MPs, called for Mr Assange to be extradited to Sweden so that the "formal investigation into an allegation of rape can be concluded".
He spent nearly seven years in the Ecuadorian embassy, where he sought political asylum in 2012 when he failed in his legal battle against extradition to Sweden, where he was wanted over two separate allegations - one of rape and one of molestation.
"It is therefore of grave concern to us that it appears that the Swedish authorities were not aware of the plans made to arrest Mr Assange yesterday in London, and we would welcome clarity as to what action the United Kingdom authorities took to ensure that the Swedish prosecutors were informed in advance of this decision", it adds.
His separate extradition case on charges of computer hacking is set to be next heard by video-link at Westminster Magistrates' Court on May 2.
Those charges were dropped in May 2017, but Swedish prosecutors say they are now re-examining the case after a request by one of the alleged victim's lawyers.
He was holed up inside the Ecuadorian embassy in London for almost seven years.
Assange denies the Swedish allegations, which have a limitation period that expires in August 2020.
The Ecuadorian government said Assange, who was thrown out of the embassy after his diplomatic asylum was withdrawn by the country's president, Lenin Moreno, had paid for his own expenses since the last December.
The 47-year-old Assange has extreme shoulder pain and tooth pain, Hrafnsson said. Regardless, Assange will have to serve any prison sentence for skipping bail in 2012 for the sexual assault and rape allegations, which a judge will determine at a later date.
Should the extradition of Mr Assange go through, it would set a awful precedent, eroding the progress made by press freedom campaigners to democratise access to information and its attendant redistribution of power to the people. The battle between Assange and the USA government was always going to be epic, involving concepts like free speech, journalists' rights, national interests, even treason. The single charge of computer intrusion conspiracy carries up to five years in prison, though the Justice Department can add additional charges depending on the evidence it gathers.
Assange's London lawyer Jennifer Robinson said: "He's obviously going to fight extradition and fight it hard". He feared that if he was taken to Sweden, he would be handed to the United States authorities.