No, you can't use the net to restrain targets this way.
Since the net is a weapon that you use an attack roll to hit with, then you must make an actual attack with it. If the net were an object that you threw, it would be used with the Use an Object action, like siege weapons are described. Sure, the description says "when you hit something" with the net, but you must read it within context.
Proficiency with a net allows you to add your proficiency bonus to the attack roll for any attack you make with it.
A Large or smaller creature hit by a net is restrained until it is freed.
When you use an action, bonus action, or reaction to attack with a net, you can make only one attack regardless of the number of attacks you can normally make.
It's not just throwing a curled up net at a creature; you have to attack with it.
Another thing to pay attention to is the actual spell description.
On a failed save, the object strikes the target and stops moving. When the object strikes something, the object and what it strikes each take 3d8 bludgeoning damage.
The impact always causes 3d8 points of bludgeoning damage. If you threw an axe with catapult, you wouldn't deal 1d6+STR as if you had made an attack with it, you would instead deal 3d8 bludeoning from the impact.
Whatever you throw is just a placeholder; there are no special properties for items.
enkryptor also makes a good point. 5e uses the hit terminology for a successful attack roll. In the spell description, the projectile is striking something, not hitting.