At one point the group carrying her coffin backed against a wall and almost dropped the casket, recovering it just before one end hit the ground as stun grenades detonated.
The violent scenes, which lasted only minutes, added to Palestinian outrage over Abu Akleh's killing, which has threatened to fuel violence that has surged since March.
Abu Akleh, who had covered Palestinian affairs and the Middle East for more than two decades, was shot while reporting on an Israeli raid in the occupied West Bank on Wednesday.
Palestinian authorities have described Abu Akleh's killing as an assassination by Israeli forces. Israel's government initially suggested Palestinian fire might have been to blame, but officials have also said they could not rule out it was Israeli gunfire that killed her.
Israeli police said a group of Palestinians outside the hospital, whom they described as rioters, had begun throwing stones at officers.
"The policemen were forced to act," they added.
The White House found the images disturbing, press secretary Jen Psaki told reporters, and US officials will remain in close contact with Israeli and Palestinian authorities in the aftermath of Akleh's funeral.
"Every family deserves to be able to lay their loved ones to rest in a dignified and unimpeded manner," US secretary of state Antony Blinken said.
Egypt, Qatar and Al Jazeera condemned the police's conduct. Deputy UN spokesperson Farhan Haq said the scenes were "very shocking" and the EU said it was appalled.
A few minutes after police intervened, Abu Akleh's coffin was placed in a vehicle that headed toward the Cathedral of the Annunciation of the Virgin in Jerusalem's Walled Old City, where the funeral ceremony proceeded peacefully.
Crowds of Palestinians lined the narrow alleyways of the Old City as the coffin was carried to the Mount Zion Cemetery nearby.
Her grave was covered in wreaths and the Palestinian flag draped over the grave cross as mourners surrounded it solemnly, paying tribute to Abu Akleh.
"We're here because we are screaming for justice. Justice for Shireen Abu Akleh and justice for Palestine," said one mourner, who did not want to be identified by name.