A word of unsolicited caution to ideological partisans: when arguing a point, do not base your case on a point of principle or theory that you outright reject in other circumstances. This is known as hypocrisy.
For two or three decades, many connected to the world of philanthropy and private foundations have been arguing that the tax exemption enjoyed by private foundations (and the charitable deduction granted to individual citizens) is a government subsidy. This argument is made principally, but not exclusively, by those on the liberal end of the spectrum and has even gone as far as one Congressman declaring that tax exemption is a federal "earmark" to foundations. The exemption/deduction = subsidy argument is made to justify expansive government intrusion into the working of private foundations: mandating types of spending, staff composition, etc. That is, there are two claims at work. The first is that foundations are subsidized through the tax code; the second is that this subsidy makes private philanthropy "public" and therefore subject to public (government) ownership and control.
In other areas of public policy this line of argument is known as "tax expenditure theory"--tax credits and deductions and exemptions are government "expenditures," whether in fact or in financial equivalence. (The Joint Committee on Taxation prepares an expenditure report in which it calculates the "cost" of deductions and exemptions and credits.) It is a long-running debate and many, including myself, have argued against the use of the theory to justify far-reaching public ownership or control of philanthropy. Perhaps not suprisingly, this argument has been made mostly, but again not exclusively, by those on the right.
Now, in the resurrected debate over federal funding of abortion, the ideological markers are reversed: conservatives argue that tax credits and exemptions for certain types of abortion services are really government subsidies and therefore need to be eliminated--or that this justifies expansive government restriction on those services. Those on the left, meanwhile, now say that such elements of the tax code are not subsidies and therefore cannot be made the crutch of this line of argument.
The tax expenditure/subsidy argument will likely never be resolved, but it doesn't necessarily have to be. Both sides (of course, it is really all sides or everyone because I can't even tell where the "sides" are anymore in the above arguments) can accept, I think, that an exemption acts as a subsidy, both in financial equivalence (though not perfecty given the frictions and costs of collecting taxes and then redistributing as a subsidy) and in terms of an affirmative government decision to encourage (or discourage) some activity through the tax code. (The tax code has long been a major vehicle for policymaking.) What I disagree with, and what both the left and the right are now doing, is taking this one step further and arguing that this equivalence or this government "decision" justifies further restrictions because that "subsidized" activity is now "public." This is not a permissible line of logic: the right cannot defend philanthropic independence by cabining the tax exemption/subsidy argument and then turn around and use that exact same argument on the abortion issue; the left cannot color all private philanthropy as "public" and then turn around and say tax exemptions are actually not subsidies in the case of abortion. Everyone needs to be more honest about the assumptions underlying their arguments. Like that will ever happen.