Effect of a wire thickness affect the magnetic field
So, can I say that only current affects the magnetic field and not the
  thickness?


This is only true in relation to the external magnetic field produced "outside" a conductor. For example: a 1 mm diameter wire will produce the same flux density at a distance of (say) 100 mm from its wire centre as a 10 mm diameter wire carrying the same current.



AC considerations

With AC this is also true but, due to skin effects, the thicker wire has less inductance. Because skin effect causes current crowding at the wire surface, the same flux density will be seen at any distant point compared to a thinner wire carrying the same current.

Consider also a conducting tube: if we compared a solid 10 mm diameter wire carrying with a 10 mm diameter hollow tube then, for the same current, the external flux density fields will be the same at any distant point.



Example - Coaxial conductors

This is admirably demonstrated by a piece of coax wire. The inner wire is thin and carries the same magnitude current as the outer shield but, the current directions are opposite and so, we expect fluxes from each to cancel. And this is what we get: the magnetic field produced by the inner wire current is totally cancelled by the returning current flowing back through the shield. This field cancellation occurs anywhere outside the coax cable and proves the point. 

Outer shield inductance: -

Between inner and outer shield there is a magnetic flux and, its presence has an interesting knock-on effect - because any current tube (aka coax shield) cannot produce flux internally, all the flux within the coax cable must come from the inner conductor. Given that the flux external to the shield is cancelled to zero, we can rightly say that the shield has zero inductance.