Did Moscow call itself third Rome because it became the center of the Catholic Church?

It is called so because "Two Romes fell". It means the Roman Catholic Church went through many reforms and changed a lot of key aspects.

Bysantian Constantinople got the Second Rome status after the First Rome "spiritually" fell and kept the original Christianity untill it fell too but physically - it went to Turkey.

So at that moment Moscow (which was the proximate successor of Bysantium) had to take the role of the new centre of the Orthodox Catholic Church.

Moscow also stands on seven hills just like Rome and Constantinople.