Going out into the cold weather even after sweating and then going out into the cold cannot make you sick.
You can only get sick if you get infected with a virus such as the cold virus or flu virus and if that virus is not around then you cannot get sick with any virus.
However you might get hypothermia from the cold if you're sweating and then go out into the cold so in a way you could get sick but not directly from the sweat and cold.
The hypothermia could lead to other sickness though so it's best to clean the sweat off of you and dry off first before going out into the cold weather just to avoid getting hypothermia.