I tried to open your attachment but you don't have any of the other parts included (part 1, part 2, part 3), so the assembly opens with no components.
I've attached a file with a cone and two spheres, which is what I think you're looking for.
There are different kinds of constraints that you can choose from, the third option is a "tangent" constraint, which is to curved surfaces what the "mate" constraint is for flat surfaces.
That said, you'll generally find it difficult to constrain curved surfaces such as spheres. Typically what I'll do is go to the original part and turn on visibility on the origin planes, as I did with the cone in this assembly. Then those planes become visible in the assembly and you can use those to constrain the part.
Note I did NOT ground the cone, but rather constrained the cone to the origin planes of the assembly. I find this extremely useful in the work that I do where I take CAD models and export the STL files for use in MATLAB and other programs. Constraining to the origin planes gives me a way to output a consistently oriented part instead of the generally random results from just grounding the "base" part.
Once you've constrained to the origin planes you can go back to the original part and uncheck the "visible" option on those planes.
