Osmond said he doesn't plan to try to bring the issue back next session.
Marina Lowe, legislative affairs director for the American Civil Liberties Union, said she is relieved Osmond dropped his bill.
"I'm pleased he's not going that direction," she said. "It's something that has far-reaching impact, so I'm glad it's not being brought up at this stage of the session."
After a meeting with the Utah Eagle Forum Family Action Coalition last week, Osmond agreed to open a bill file titled "Joint Resolution on Human Life."
Osmond would not say where he planned to draw the line on when life begins, but opened the file to begin a dialogue.
"There's no doubt we're trying to define when life begins, when a person becomes a person," he said when he began the effort.
If approved by two-thirds of both chambers it could have been put on the ballot in November as a constitutional amendment.
But in the face of an outpouring of opposition, Osmond apologized to his Senate colleagues for distracting from the work they were doing.
Personhood amendments seeking to define life as beginning at conception have failed in Mississippi and Colorado after divisive and hard-fought campaigns. An additional dozen states are looking at similar amendments this year.
"I'm glad our state doesn't have to go through that," Lowe said.