Well, being a believer but not a Catholic, I also have wrestled with this issue, and have not come upon a satisfactory answer. But I would like to make a couple of observations.
First, Irish is not the only language to employ references to deity in their greetings and salutations, or their goodbyes for that matter. In Bavarian German, "Gruss Gott!" is used for hello, meaning "God is great." In Spanish, good-bye is "adios", literally "to God" (or God be with you), in French it's "adieu" and has the same meaning. So, are we going to purge God and religion from all languages across the globe? There is a certain wisdom in the "get over it" philosophy which spans several cultures as you can see. I don't feel comfortable saying "Dia duit" as "hello," but I've never had a problem saying "adios" to someone when trying to speak Spanish, so it's really a bit of hypocrisy on my part.
Might I venture to suggest that the exchange "Dia duit" followed by "Dia duit 's Muire" (and the less common but nonetheless published "Dia duit 's Muire, agus Pádraig!") is objectionable to many because it is so definitively Catholic. I would feel far less annoyed or put upon or whatever to reply simply "Dia duit" and leave it at that. This has the advantage of employing the basic idiom of Gaelge without having to resort to the "Catholicized" element. I know the Irish feel strongly (at least many do) about their religion; I should know, I spent two years there as a Mormon missionary. The idea of adding "'s Muire" seems to stem from an apparent need found in several places in the language to reply in some form different from the source statement. Thus the reply to "Slán agat" is "Slán leat". There are other examples, but it's almost like the one making the reply has to show some one-upsmanship in the response. I think that should be easier to let go of and just say "Dia duit" in reply and thus not feel so forced into a corner.
Finally, might I suggest that there has been an equal (at least) amount of fighting and warfare done in the name of non-religion as in the name of religion. We're all aware of the Crusades and many other confrontations where religion was front and center, including the painful years of "The Troubles" in Northern Ireland, which I experienced first hand between 1973-75. It has been suggested by some that the heart of the National Socialist (NAZI) movement in Germany was an attempt to eventually codify the Darwinian notion of "selection", breeding people the way that we breed animals and feeling no guilt at getting rid of those deemed "inferior." "Mein Kampf" lends a lot of credence to that assertion, so to that extent, it could be argued that, regardless of the wars over religion and belief over the generations, the worst conflict in Earth's history could be attributed to Darwinism. "Darwin duit"?! I, for one, would feel even *more* uncomfortable with that!
No culture or language is perfect. In the United States, English is such a hybrid language that Shakespear would roll over in his grave, and the drive to "political correctness" has resulted in many pathetic casualties, and significant backlash. I like the idea of saying, "Get over it" in certain cases. To a degree, this is one, so long as I'm not forced to accept or use the "Catholic" elements that would make me feel very uncomfortable. By the same token, I'm not willing to impose some other construct that may be equally if not more objectionable.
Slán!
Dale D