Your passport is not a residence permit
The Schengen 90/180 rule counts days across the whole area, not per country. Buying a house changes nothing about it. If you want longer, you apply for a national long-stay visa: Italy's elective residence visa, Spain's non-lucrative or digital nomad visa, Portugal's D7, France's long-stay visitor visa, or Greece's financially independent person visa. All of them want proof of stable income and private health cover.
Note also that the EU's EES entry-exit system and ETIAS authorisation change how short stays are recorded and pre-approved. Neither restricts buying property, but both affect travel planning.
Banking and the extra paperwork Americans get asked for
FATCA makes some European banks reluctant to onboard US persons. It is not a wall, but expect more documentation and a longer account opening. Start it before you need it, because you generally cannot complete a purchase without a local account for the transfer and the utilities.
On the US side: FBAR filing if your foreign accounts exceed $10,000 combined at any point in the year, and Form 8938 above higher thresholds. Owning the property itself is not reportable, but the bank account funding it is.
Currency is a real line item
On a 400,000 EUR purchase, a two-cent move in EUR/USD is roughly eight thousand dollars. Use a specialist FX provider rather than a wire from your US bank, and consider a forward contract to fix the rate between the preliminary contract and completion, which can be several months apart.
Financing from abroad
Some European banks lend to non-resident Americans, typically at 50-70% loan to value with income verification and life insurance requirements. Rates are often lower than US equivalents but underwriting is slower and paperwork-heavy. Many buyers pay cash and refinance later, or borrow against US assets instead.
Taxes on both sides
The US taxes citizens on worldwide income wherever they live. Rental income from a European property is reportable, as is a capital gain on sale, with foreign tax credits usually preventing double taxation. Each of the five countries has a tax treaty with the US, but treaty interaction with local wealth or inheritance rules varies. Use an accountant who handles both sides.
Figures are typical ranges rather than quotes. Confirm your own numbers with a local notary, lawyer or tax adviser before you commit.