Is there anything in Tap Magic cutting fluid that suggests that I should avoid using it on my Arkansas sharpening stone?
I see that Tap Magic make more than one product but I'll answer generically.
Regardless of the specific ingredients given what Tap Magic is for I doubt it contains anything that could harm Arkansas stones given they are very tough to begin with. So I'd say it's safe enough to try out at least, but I wouldn't recommend it in the long term especially.
If it does work well as a honing fluid it would join a laundry list of liquids that someone somewhere has been happy to hone with, but you might want to use something that's more skin-friendly. At least one of the Tap Magic fluids is classed by the maker as a skin irritant so I would say on that front alone you should look at an alternative, unless you're routinely in the habit of honing with gloves on.
Since a number of cheap and completely safe liquids can be used as honing fluid (including soapy water, mineral oil/liquid paraffin and baby oil) I'd say there's a strong argument not to use Tap Magic for this purpose.