The Nashville Predators selected Fabbro, a defenseman who likes to join the attack, in the first round (No. 17) of the 2016 NHL Draft. Fabbro was coming off a 14-goal, 67-point season with Penticton of the British Columbia Hockey League, when he was named the league's best defenseman.

Fabbro had played in the BCHL to preserve his NCAA eligibility. He played the next three seasons at Boston University, where he had 80 points (22 goals, 58 assists) in 112 games, as well as helping Canada finish first at the 2018 IIHF World Championship, before signing with the Predators on March 27, 2019. He made his NHL debut against the Columbus Blue Jackets four nights later and scored his first goal one week later against the Chicago Blackhawks on April 6, 2019. Fabbro had one assist during Nashville's six-game loss to the Dallas Stars in the first round of the Stanley Cup Playoffs.

After the Predators were eliminated, Fabbro helped Canada to a second-place finish at the IIHF World Championship. He stepped into a regular role on Nashville's defense during the 2019-20 season.

Fabbro signed a one-year, $2.5 million contract with the Predators on March 3, 2023, toward the end of a two-year, $4.8 million contract he signed Aug. 5, 2021. He was claimed by the Blue Jackets off waivers from the Predators on Nov. 10, 2024, and had 26 points (nine goals, 17 assists) in 62 games to earn a four-year, $16,5 million contract he signed with Columbus on June 29, 2025.