Factual error: The type of surgery shown made by the machine transversely cuts the rectus abdominis muscles. Besides not being a standard surgical incision in order to "remove a foreign body" as stated, it seriously damages important muscles of the abdominal wall making any immediate active mobilization of the trunk virtually impossible, even under strong analgesia.
Factual error: Carbon dating cannot be used without knowing the amount of C14 in the atmosphere and its evolution in history. Using this method just after arriving at an alien planet is impossible.
Revealing mistake: Each crew member has a monitor on their spacesuits that they use whenever they are exploring the caves. Whenever the monitor is shown, each crew member's individual heartbeat is shown in the lower left corner of the monitor. But no matter which heartbeat is shown, every time a heartbeat on the monitors is shown, it is always the exact same heartbeat, despite the various changes in panic and stress throughout the movie.
Character mistake: In the first scene inside the Prometheus, after she comes out of deep sleep, Vickers asks David for a robe, and asks "how long?" meaning how long have they been asleep. David, a robot, answers "2 years, 4 months, 18 days, 36 hours, 15 minutes." The actual statement from this highly precise robot character would have been "2 years, 4 months, 19 days, 12 hours, 15 minutes."
Continuity mistake: The 2 who are left behind to spend the first night there, due to the storm, are sprayed with a horrible spray when one cuts the alien creature as it wraps around the second's arm. They both get spray on their faceplates, but in the next shot the one that hasn't got it wrapped around its arm has no spray on his faceplate.
Factual error: In the scene in the ship where Charlie exhibits the ancient artifacts that depicts the planetary system they are traveling to, he shows an artifact from Sumer, Babylonia, and Mesopotamia, and refers to these as "unconnected" cultures. In fact, the terms Sumer and Babylonia both refer to ancient Iraq, also known as Mesopotamia. The terms Sumerian and Babylonian are generally used to refer to different time periods in Mesopotamian history. These three terms refer to the same culture, and are not at all unconnected as Charlie said.
Other mistake: In the scene with the crashed escape pod, everything inside is upended with books and stuff strewn everywhere. Yet there is a bottle of Vodka sitting perfectly upright on the bar in the background.
Factual error: As the ship lands the ship's voice states "temperature at 2.725 K." At that temperature, only helium would be a gas. All others would be frozen solid, thus probably no atmosphere. In a virtual vacuum. It would be virtually impossible for a breathable atmosphere to be present within the structure. This temperature is incompatible with life.
Continuity mistake: During the final scenes, after the ships have collided and crashed to the surface, Elizabeth finds David's detached head inside the alien ship. She then takes the head with her to a ledge where she is about to lower David's body to the ground. In both places, David's hair keeps changing between shots: messy/neat/very messy, with no logic whatsoever.
Continuity mistake: When Shaw is on the planet surface towards the end of the film, she isn't wearing any gloves. Later on, she suddenly has gloves again.
Continuity mistake: When Vickers enters Peter Weyland's room to speak to him just before he embarks to see the engineer, her hands are clasped behind her. They remain that way in each shot where she is seen from behind. But every time the camera angle changes and we see Vickers from the front, her hands are suddenly clasped in front of her. This switching occurs repeatedly throughout the scene.
Revealing mistake: When Fifeld and Millburn find the dead bodies, just after Millburn says "This thing opened up from the inside. Looks like it exploded", the shot cuts to Fifeld flashing his torch over the corpses. During this shot, a corpse's arm to the right of Fifeld is shown wobbling around.
Factual error: In the scene showing the two left-behind crewmen as they discover the snake-like alien, one of them reports to the ship that the creature is "30-40 inches long." The scientific community worldwide has been using the metric system for decades and it would seem highly unlikely a trained member of a starship crew would revert to such an ancient system of measurement.
Continuity mistake: Prometheus is flown into the alien spacecraft, presumably to disable it from flying. However, after the alien craft has crashed and rolled over Meredith Vickers, the ship is facing face down on the planet, giving us a full shot of the back of the ship. There is no damage to the back of the bulkhead whatsoever, even after the bright explosion in the atmosphere. (01:43:00)