Nassib received an athletic scholarship to attend Syracuse University, where he played for the Syracuse Orange football team from 2008 to 2012. He redshirted the 2008 season, sitting on the bench behind Cameron Dantley. Nassib was named as Syracuse's new starting quarterback during 2009 spring practices, jumping over Dantley and Andrew Robinson. After just one week of fall practice, however, head coach Doug Marrone named Duke transfer Greg Paulus as the team's 2009 starter. Backing up Paulus, Nassib played 10 games during the 2009 season where he played a role in specific offensive packages against Minnesota and Northwestern. Against Penn State, he completed 4-of-5 passes for 30 yards, including his first completion on his first career pass. He ended the 2009 season throwing for 422 yards and three touchdowns with a 116.69 QB rating.
With the departure of Paulus, Ryan moved up the depth chart as he became the starter for the Orange for the 2010 season. Nassib threw at least one touchdown pass in eleven games as he tied for fourth on Syracuse's single-season record list with 19 touchdown passes. At Akron, Nassib led Syracuse to its first season-opening victory since 2003 as he threw for 229 yards and two touchdowns on 17 of 27 completions. Against Washington, he completed 17 of 35 passes for 202 yards and a touchdown as he also ran for 39 yards and one touchdown on seven carries. His 28-yard rushing touchdown was the first of his career. Against Maine, Nassib set a school record with five touchdown passes. Nassib piloted a 98-yard scoring drive in the fourth quarter and a three-yard touchdown completion to Marcus Sales to win the game against South Florida. The Orange finished the 2010 season with a 7–5 record and a Pinstripe Bowl win with Ryan completing 13–21 pass attempts for 239 yards and three touchdowns, including a career long 52-yard completion.
In the 2011 season, he set the Syracuse single-season records for completion (259) and passing yards (2,685) and tied the mark for touchdown passes in one season (22). Down 15 points in the season opener against Wake Forest, Nassib led the Orange to a comeback win in overtime with a 4-yard touchdown pass to Van Chew. He finished the game completing 20 of 28 passes and 178 yards, including 8 consecutive passes in the fourth quarter and overtime. Nassib earned Big East Weekly Honor Roll recognition with a 25 for 37, 230 yard, one touchdown performance against USC. Nassib was once again named to the Big East Weekly Honor Roll against Toledo as he completed 16 of 24 passes for 213 yards and two touchdowns in an overtime victory. Although Syracuse ended the 2011 season with a disappointing 5–7 record, Nassib had a decent season as he had a .624 pass completion percentage record, 2,724 yards of total offense ranks second and 223.8 average passing yards.
Nassib started the 2012 season strong against Northwestern completing 45-of-66 passes for 482 yards and four touchdown passes as the Orange lost by a score of 41–42. His performance earned him Big East Offensive Player of the Week honors as he set Big East records for pass completions, pass attempts, and passing yards. Against the undefeated Louisville Cardinals in his last game at the Carrier Dome, Nassib led the Orange to a 45–26 victory as he completed 15 of 23 passes for 246 yards and 3 touchdown passes. With wins against Missouri and Temple, the Orange had a 7–5 record and were bowl-eligible. Syracuse beat West Virginia 38–14 in the Pinstripe Bowl. In his final college game, Nassib completed 12 of 24 passing attempts for 134 yards, 1 interception, and 2 touchdown passes. Nassib was named the National Football Foundation National Scholar-Athlete Award winner presented by Fidelity Investments and he also made the 2012 Capital One Academic All-District Team selection. The senior signal was named a to Pro Football Weekly's 2012 All-America Team. Nassib finished his career with Syracuse breaking the school total passing yards record (9,060), pass completions (780), and passing yards per game (201.3). Nassib became the fourth Big East quarterback to pass for more than 9,000 yards.
Nassib was selected with the 110th overall pick in the fourth round of the 2013 NFL Draft by the New York Giants. He was the primary backup to starting quarterback Eli Manning.
Nassib made his NFL debut in the 2014 NFL season in a 45-14 win over the Washington Redskins in Week 4.
On December 27, 2015, in a 49-17 loss to the Minnesota Vikings, Nassib threw his first NFL touchdown pass to wide receiver Myles White in the fourth quarter. He finished the game passing 5-for-5, 68 yards, and one touchdown, along with a 158.3 perfect passer rating.
Nassib was placed on injured reserve on December 20, 2016.
In the offseason, Nassib was released by the Giants.
On June 12, 2017, Nassib signed a contract with the New Orleans Saints. He was released on September 2, 2017.
Nassib is married to Madeline Paolantonio. He is the son of Mary and Gilbert Nassib. His father played tight end at the University of Delaware. He has four siblings, two younger brothers, John and Carl, and two sisters, Carey and Paige. Carl was drafted in the third round of the 2016 NFL Draft by the Cleveland Browns while John is a senior defensive end for the University of Delaware.