When a mass moves towards a bigger mass at a constant velocity, they will be attracted to each other such that the smaller mass will revolve around the bigger mass with the same speed as before.
However, if the above case is true, the total angular momentum of the system will increase from zero to non-zero. This is clearly a violation of conservation of angular momentum.
Edit: when I am talking about this, I don't mean that two mass are moving towards each other. I mean for instance there is a stationary planet and a mass such as an asteroid move horizontally above the planet at a constant speed. The asteroid is moving past the planet without touching it. This asteroid will then be attracted to the planet gravitational pull, resulting in it turning direction and begin to revolve around the planet.
This is just hypothetical and I don't see how this is impossible.
Best Answer
To begin with, it is important to clarify the scenario. When two masses are attracted to each other, regardless of their magnitudes, the gravitational force (and hence acceleration) vector acting on each is in the direction of the other. If two masses were sitting motionless in a vacuum, they would accelerate toward each other in a straight line until such time that they collided.
In order for masses to fall into an orbit, they must have some component velocity that is perpendicular to their gravitational acceleration vector. The total velocity of each mass at any point in time is the sum of its velocity in the direction of its acceleration (that is, in the direction of the other mass) and its perpendicular component. It is the perpendicular component of their velocities which gives the system its angular momentum. The masses revolve around some point between their two centres.
Thus, in the scenario which you describe, the system begins and ends with the same non-zero angular momentum.