The incident happened at 9:14 a.m. (4:14 a.m. ET) outside a sports center.
All six pedestrians, including three children, were taken to a hospital. Police say two children remain in pediatric intensive care and one adult is receiving treatment in the trauma high dependency unit.
The other patients have suffered serious but not life-threatening injuries, authorities said.
Police confirmed that a 42-year-old woman was arrested and added that they were not pursuing anyone else.
"What we have established is that a 42-year-old female has been celebrating Eid with her family, she then got into her car and has collided with six people in the crowd," Assistant Chief Constable Darren Best said in a statement.
"We have no information to suggest this is terror-related, however, this is a serious collision with multiple casualties and extensive inquiries are ongoing to establish the circumstances around this tragic incident."
Authorities said extra officers have been deployed to patrol in Newcastle and surrounding communities to "reassure members of the public."
A statement from Newcastle Central Mosque on its Facebook page
said that the incident took place immediately after prayers.
Eid al-Fitr marks the end of the holy fasting month of Ramadan.
Islam's lunar calendar means that the time of Eid can vary in different countries, depending on the sighting of the new moon. Events are taking place across the UK, with the largest of those, Eid in the Park
, happening in Birmingham.
The incident comes less than a week after one man died and others were injured when a man plowed a van into a crowd of worshipers leaving Ramadan prayers in Finsbury Park, London. Authorities have charged a man with terrorism-related murder and attempted murder.