Judge Alito's nomination was approved by the Senate Judiciary Committee by a 10-8 party line vote (per How Appealing). One Democratic Senator has said that he will vote for Alito. There are 55 Republicans in the Senate. In order to win on floor vote, six of those Republicans need to oppose Alito, and all remaining Senators must oppose Alito. This seems very unlikely.
Alternately, Democrats could filibuster the nomination (something Republicans have done before themselves in recent history on a judicial nomination), and no more than three of the remaining Senators (the 43 other Democrats and independent Jim Jeffords) could oppose cloture. If Democrats lose the cloture vote, Justice Alito is confirmed and nothing really unusual will have happened.
If Democrats win the cloture vote, then the Republicans have a stark choice whose outcome is uncertain. If the Republicans fail to marshall a majority of Senators to abolish the filibuster for judicial nominees entirely, the Alito nomination, George W. Bush's second nominee for this post will be defeated (actually, his third, if you count Justice Roberts whose nomination was withdrawn so he could be nominated for Rehnquist's post) and he will likely feel compelled to appoint a candidate both more moderate than Alito and more conservative, or at least experienced, that Harriet Miers, who withdrew her nomination, and there will be no constitutional crisis.
But, Republicans could still confirm Judge Alito if they abolished the filibuster for judicial nominations, and their loyalty to their base, which is far greater than that of Democratic Senators, makes this a close question. If Republicans vote to abolish the filibuster (I doubt that any non-Republicans would join them in this vote, and I doubt a court would intervene if the Republicans did so, since the Senate is the judge of its own rules), it would have immense long term consequences for both parties.
The Republicans would, in my opinion, pay a higher political price for their decision in the long term than the Democrats, both at the polls for showing a lack of fair play, and when they are in the minority some day and don't have a filibuster for judicial appointments, and also can't count on a filibuster in high stakes legislative fights as they established the precedent for eliminating it in any situation. In contrast, the Democrats have little to lose. The need to abolish the filibuster to confirm Judge Alito's nomination would also tarnish newly confirmed Justice Alito in the eyes of the public, even if he is confirmed as a result, and would also further undermine the legitimacy of the U.S. Supreme Court in the eyes of the public (already suffering from the blow to its credibility arising from the Bush v. Gore case).
The Alito nomination is the most important nomination for the Democrats to oppose in decades, at least, and no nominations more important to oppose are likely to arise for decades to come. This is because he is a hard core conservative who will move the Supreme Court decisively to the right. The Democrats are idiots if they don't try to filibuster this nomination. Even if desertions on their own side prevent them from prevailing on a cloture vote, members of their base, like me, will not hate them forever if they at least try and a few traitor Democrats foil their attempt. Alas, I don't have a high opinion of the intelligence of the Democratic party leadership in the Senate on issue like this one.