There are perhaps just four countries in the Middle East which can by any definition be called democratic: Iran, Iraq, Israel and Lebanon.  Palestine does not yet exist as a country, but if we add the Palestinian Authority, that makes five. 
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Of these, Iran is openly hostile to the United States; the Palestinians have elected a party to power which has been fighting a deadly war with America’s closest ally; and in Lebanon, one of the major political participants is Hizballah, still regarded by the US as a highly dangerous and hostile terrorist organisation.  Iraq’s future is completely unpredictable.  Only Israel can be seen as an unmixed blessing for the US, and even there, it is arguable that the US gains far less from its relationship with Israel than it gives.
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Contrast the situation with the other countries, undemocratic to a man.  America’s greatest supporters for many years have been Egypt, Saudi Arabia, Jordan, the Gulf States (all of them), and all the North African states except Libya. 
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In fact in recent years, the foes of the US in the region have been limited to Iran (after the advent of democracy), Iraq, Syria, and Libya.  Yemen supported Iraq in the first Gulf War, but is now, if not exactly in the Western camp, at least benignly neutral. Given that Libya now appears to be no longer a threat and Iraq is, for however long that may last, an ally, that leaves just Syria as the one undemocratic country in the region which represents any threat to the US.  The state of the Syrian armed forces means that the threat is confined to support to terrorist or insurgency groups in Iraq.  Serious though this is, it is a minor threat compared with that from Iran.
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Yet the Americans are pushing, or claim to be pushing, for democracy.  If their wish were to be granted, what would be the likely outcome?  When free elections were permitted in Algeria, an Islamic slate was elected that was prevented from gaining power by the military, to Washington’s approval.  It is safe to predict that many countries would elect Islamic governments, if allowed to do so, that would be less friendly to the US than the present regimes, and these would include keystone states such as Egypt, Algeria and Saudi Arabia. 
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So why do the Americans claim to be aiming for an outcome that would, on balance, damage their interests?  Cynics might argue that they are simply being hypocritical, and have no intention of pushing poweful allies such as Egypt towards anything that would threaten the status quo.  In Palestine, where they had no means of preventing elections and did not predict a Hamas victory, the US is refusing to deal with the new representatives until they agree to recognise Israel and declare an end to violence: in other words, they are effectively refusing to recognise the will of the Palestinian people. 
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But things are seldom that simple.  The rhetoric of democracy has a momentum of its own.  It is possible that the American leadership has not really thought through, clearly and realistically, the consequences of a consistent policy of democratisation in the region; or maybe they think that regimes such as that of Mubarak in Egypt will, under steady pressure, give ground to the forces of change and evolve towards greater freedom without jeopardising the regime or its fundamental alignment with the US.  Sadly, the history of the Middle East shows that fundamental change comes violently and catastrophically, and not via some cosy evolutionary process in which nobody gets hurt and everyone must have prizes.
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So what do the American people make of all this?  Perhaps they still believe, despite the evidence, that America would benefit from democracy in the region.  Readers of Alice through the Looking Glass will recall that the White Queen could believe as many as ten impossible things before breakfast.  But the book was a dream; how long will it be before reality intrudes into the American consciousness?