Revealing mistake: When Salt sits on the helicopter there are two of the same cuts with Salt looking through the heli's window with yawning guy in the background. Cuts split by the view of Washington DC.
Revealing mistake: During the scene where Salt spreads her belongings on the bed of her hotel room after the opening chase, you can see in the close-up shot that the primers of the bullets are missing. This means that the bullets are lacking the necessary components (both the primer and the gunpowder) to actually be fired.
Revealing mistake: In the White House after they have captured Salt and are walking her chained to be transferred, they meet Ted up the stairs and she chokes him with the chains. Liev Schreiber is replaced by a double with a different hair cut. Ted's hair is short while the double's is long.
Answer: They have to at least investigate that kind of an accusation, and when Salt so vigorously resists arrest it basically confirms, in their minds, that she is in fact a spy.
Phixius ★
I actually kind of find it shocking that they would be willing to believe a guy who would do or say anything to save himself and since Salt was with the agency a long time, they could have at least given her the benefit of the doubt.