"Why would it? Because if I have $100,000, I can buy a $100,000 untaxed existing home, or I can buy a taxed new home and need an extra $23,000 in tax (or alternatively, I can buy a lot less house and pay the tax and house out of the $100,000)."
Apples to oranges. Under the FairTax, corporate income and payroll taxes will have been eliminated, along with their enormous compliance costs. Therefore, a house that would have cost $100K in price will be significantly less under the FairTax. When you add the sales tax onto the reduced price, you are back to about the $100K level. The biggest difference is that the tax burden will be explicitly broken out, rather than buried in the purchase price as before.
Apples to oranges? No. This is not a before v. after scenario. I'm talking about a day after you have succeeded in killing the income tax and instituting the 'fair tax.' At that point, I am making a choice between an existing home or new construction.
Taxing the new construction but not the existing home provides a significant disincentive to new construction. I'm guessing you don't work in the trades, or in logging, or manufacturing of construction goods....