Staffordshire Police said the man, who had previously been released on bail, had been re-arrested after a body was discovered near Caverswall on Saturday.
Two other men, aged 28 and 60, have been arrested on suspicion of assisting an offender, the force said.
Ms Eastwood, 28, was last seen in uniform leaving work at Royal Stoke Hospital at 7.45am eight days earlier on Friday July 27.
Forensic teams are at the scene and will remain there for some time, while specialists officers support the midwife’s family.
Police said the body has yet to be formally identified and a post-mortem examination will take place in due course.
Dozens of officers and police dogs had been scouring an area near Tickhill Lane, using large sticks to search bushes.
Nigel Potts, who owns neighbouring Newhill Farm, said he had witnessed a large police presence at a nearby disused quarry on Saturday morning.
“It seems that that’s where the main focus of the search is. They’ve been down here for three days,” the 53-year-old said.
He added that officers asked to view footage from his CCTV camera on the night the midwife disappeared, but his recordings did not go back that far.
Ms Eastwood’s colleagues raised concerns at 7.20pm on the same day she was last seen after she failed to show up for her next shift.
The force said neighbours reported hearing a scream in the vicinity of Ms Eastwood’s house at about 2pm on the day she went missing.
Detective Superintendent Simon Duffy, from the force’s Major and Organised Crime Department, said: “The discovery today is not what anyone wanted, and this is now a homicide investigation.
“A 32-year-old man who was previously released on conditional bail has now been re-arrested for murder, and remains in custody.
“Two other men, aged 28 and 60, have been arrested on suspicion of assisting an offender.”
He added: “I would like to thank the public for their support, and anyone with any further information is asked to contact us on 101.”
Ms Eastwood’s sister, Gemma Eastwood, had made a tearful appeal for help in tracing Samantha on Friday, saying: “We miss you very much, we need you home where you belong.
“If she’s out there, or anyone knows where she is, please get in touch.
“Get her home.”
Liz Rix, Chief Nurse, University Hospitals of North Midlands NHS Trust, said: “It is with deep sadness that we announce that Staffordshire Police have discovered a body, yet to be formally identified, in the search for our friend and colleague Samantha Eastwood.
“Our thoughts are with her family and friends during this time. We would like to thank Staffordshire Police for their dedication and support this week.”