Smith was "cold-cocked ... sucker-punched, whatever you want to call it," coach Todd Bowles said in a pre-practice briefing with reporters.
Enemkpali, a sixth-round pick in 2014, was released immediately by the team.
Smith, who was the presumptive opening-day starter, will have surgery, according to Bowles.
A clearly annoyed Bowles said Enemkpali punched Smith over "a very childish matter ... something two sixth-graders could've handled." This means veteran backup Ryan Fitzpatrick will start the season as the quarterback. Smith, entering his third season, was having an impressive camp. He didn't throw his first interceptions until Monday. Six weeks brings him back possibly as early as Sept. 20 (a couple days shy of six weeks which is Week 2), but up to 10 weeks could keep him out well into October.