There is nothing stable about countries that are governed by tyranny. Conservatives like you tend to make the mistake of assuming that maintaining the status quo of friendly foreign governments preserves "stability" while liberals tend to assume that rule by popular vote will make foreign countries more like us.
What you both forget is that what gives America its strength and prosperity is our traditional (albeit eroding) respect for responsible liberty. You can have majoritarian democracy without responsible liberty (eg. South Africa) and you can have responsible liberty in a monarchy. What the Middle East has traditionally lacked is an ethos which respects responsible liberty, and that is what we should be supporting there, not "stability" or "democracy".
Is having one's basic rights being decided by the masses something that is desirable? If the Saudis had something comparable to our Constitution, having a monarchy would not be so bad.
Unfortunately, our nation has gone from being a democratic republic to a republican democracy, whereas 535 people are chosen to vote willy-nilly on what our rights should be.
While there is no shortage of countries ruled by tyranny who have tumbled, your statement is simply wrong. Would you describe China as unstable today? Were the Balkans more stable under Communist rule than before or after?
Preserving friendly foreign governments is in our national interest, but that doesn't mean we don't seek to cause them to move toward responsible liberty for their people. But the circumstances have to be appropriate.
Pakistan provides an excellent example. While we have always applauded their democratic principles, we are actually quite pleased that General Musharraf is in absolute control of that government right now. He has provided remarkable stability at a time when the whole region could have exploded if bin Laden had achieved his goals.
If that makes me "mistaken" in your book, I can live with it.