Are only Catholics/Christians allowed to visit the Vatican?
If not, can the non-Catholics access the same areas as Catholics?
Vatican City is certainly open to visit for tourists at large; as you perhaps know, there is no actual boundary between Rome and Vatican City, meaning that nobody checks the papers of those who move from one to the other.
However, when I first read your question, I thought that you meant St. Peter's Cathedral, which takes up so much of Vatican City and which characterises it from a visual point of view with its huge dome. If this is the case, not only do rules apply for proper clothing, but there are also areas inside the church which are considered off limits for all those who are not there for devotional reasons. There are portions of the church which are cordoned off and which you may enter only if you intend to pray. Once again, nobody will ask you for proof of your being a Catholic, but you may be asked to leave if, instead of praying, you take photos or perform other "touristy" actions.
With 109 acres (44 hectares) within its walls, the Vatican is easily traveled by foot; however, most of this area is inaccessible to tourists.
No reference about Anglicans, Muslims or any other - just tourists. Any where that a Catholic can get in, you can too, regardless of your background. And also importantly, regardless of your gender - a recent change!
Since Vatican City is a Papal state, as such respect and reverence to the Roman Catholic Church and its practices and doctrine is encouraged.
Sleeveless shirts and short pants or skirts are not permitted within the border of the Vatican.
Some areas typically off-limits can be applied for access to them, including the famous Vatican Secret Archives.
Qualified scholars from institutions of higher education pursuing scientific researches, with an adequate knowledge of archival research, could apply for an entry card. Scholars need an introductory letter by either a recognized institute of research or by a suitably qualified person in the field of historical research. Applicants need to specify their personal data (name, address etc.) as well as the purpose of their research. Undergraduate students are not admitted.
Getting into Vatican is pretty straightforward. There are no border checks. Problems may arise when you want to enter St. Peters Basilica. However, religious beliefs don't matter. I am not a Catholic and I have already visited Vatican and St. Peter's basilica twice. Nobody asked me about my beliefs.
I can think about three things which are eliminatory:
Also note that for visiting places other than St. Peter's square, St. Peter's Basilica, and the museums, you need special arrangements.