Unless the public schools are really bad, I'd rather stick them for the socialization. When you get out in the real world, you don't get to choose your bosses or co-workers. I feel public schools provide greater exposure to a greater diversity of the general population, and you learn how to adapt and handle different types of people better. Don't underestimate the value of people skills.
Private schools or what?
Well not having gone to a private religious-based school (K-12) I don't have any personal insight into the curriculum or situation, but it seems to me that the bottom line is that you should just check out what schools are an option for you, audit some of the classes if that is permissible and essentially investigate the quality of education that your child would be getting.
With the religious schools you might discuss your concerns with the director or principal or whoever. I suspect that your concerns are not uncommon as many non-religious parents send their children to private religious schools.
Having said the above, I would not object to having my child attend a religious private school and in fact have greenlighted my current daycare provider to my 3-yr old daughter who gives some instruction along the lines of her bible (Christian) views and has even taught my daughter to say Grace among other things.
The reason I would have no qualms about it are two-fold:
First, learning about religion is an education. Indeed it is an important facet of an well-rounded education as most of the people on the planet are religiously inclined and to ignore the entire subject area would be akin to ignoring subjects like History, Math, Science, etc.
Secondly, as with all things, you have your own job to do as a parent in instructing and educating your child - I don't have to tell you that I am sure. Thus whether your child hears questionable (or objectionable) info at school about Religion or whether it be about SEX or about Race or anything at all, it is YOUR JOB to provide correct viewpoint or your other information and to teach your child to think critically and to make judgements and arrive at conclusions themselves and ultimately to develop their own viewpoint.
Therefore, in light of this, it seems to me that your primary concern should be not on the religous aspects of the education but on the overall quality of the education. In that regard the private school may excel the public school offering or it may not.
As for social aspects or extracurricular aspects, I am afraid that I don't have the experience to know how these things might compare in the two settings.
best of luck,