There are 2 types of force being considered here:
the 'gravitational' force you mention, and the normal contact force.
If, by 'gravitational' you mean the force of the Earth pulling down on the object, then that is contactless, action at a distance.
The normal reaction is a contact force.
You also used the word 'weight', but I will assume you mean the Earth/Mass gravitational attraction here.
While that gravitational force is often referred to as 'weight', the weight of an object is normally experienced as the force it pushes down with on the surface it is resting on. That actually would be a contact force.
So, the gravitational and normal reaction forces are not of the same 'type'. One is contactless and the other is contact.
btw.
The Newton's 3rd Law reaction to the gravitational force of the Earth on that mass, is the equal and opposite force of that mass pulling up on the Earth. They are of the same type, being both 'action at a distance'.
It can occasionally get confusing in discussions of 'weight', as to exactly which force is being referred to here.
Let me know if this helps.