Born in Massapequa, New York, Nealis began playing soccer at the age of seven.[2] In his junior year of high school, he led the Massapequa Chiefs to a NYSPHSAA Class AA Championship. Nealis attended Hofstra University, where he was a member of the varsity soccer team for four years. During his senior season he was recognized as the Colonial Athletic Association Defensive Player of the Year.[3] In 2018, Nealis played five games in the PDL with Westchester Flames.
On January 11, 2019, Nealis was drafted in the second round of the 2019 MLS SuperDraft, by the New York Red Bulls.[5] After a strong preseason with the club, Nealis signed an MLS contract on February 17.[6] On February 27, 2019, Nealis made his professional and New York Red Bulls debut coming on in the second half for Aaron Long in 3–0 victory over Atlético Pantoja in the 2019 CONCACAF Champions League.[7] A few days later, on March 2, 2019, Nealis made his league debut with the club, appearing in a 1–1 draw with Columbus Crew.[8] On May 11, 2019, Nealis scored his first goal for the club in a 3–1 victory over FC Dallas.[9]
Nealis became a regular starter for New York during the 2021 season. On August 25, 2021, he signed a three-year contract extension, with an additional option year.[10] On October 23, 2021, Nealis scored the winning goal for New York in a 2–1 victory over Columbus Crew.[11]
Toward the end of the 2022 season, New York once again handed Nealis a contract extension through the 2026 season with an option for 2027.[12] Nealis was named team captain for New York for the 2023 season. On September 23, 2023, Nealis scored his first goal of the season, the go ahead goal in a 5–3 victory over D.C. United.[13]
Nealis scored the only Red Bull goal in a 2–1 loss to the LA Galaxy in MLS Cup 2024.