Icarus's gravity isn't explained in the film but we must assume some sort of artificial gravity generator other than spinning crew compartments, otherwise the mission isn't survivable. The problem is their flight path and the Sun's gravity.
A mission simulation was shown around 22 minutes into the film.
We see Icarus drive straight toward the sun just as they would on Star Trek. No futzing around with transfer orbits, they just go straight in like a skydiver, release the bomb, and accelerate straight away from the Sun. Four minutes later the bomb fires its engines and dives into the Sun. The problem with such an aggressive flight plan is that near its surface the Sun's gravity is 27 times that of Earth. Without an artificial gravity source to negate that force, just holding station above the Sun will crush everyone aboard Icarus into paste and likely pancake the spindly-looking ship against the thrust of the engines.
The only way the mission works without artificial gravity is if Icarus does a burn to inject itself into a hyperbolic or elliptic trajectory that is a close flyby of the Sun. At perihelion Icarus separates from the bomb, which does a final burn to plunge into the Sun, while Icarus coasts along the outward leg of the orbit away from the Sun. Icarus would be in free fall throughout the close part of the flyby, so the Sun's high gravity would not be a hazard.