Why didn't America finish Saddam Hussein after the first Gulf War?

George Bush Sr. (a former oilman) was mainly concerned about the oil of Kuwait, and worse, Saudi Arabia, falling permanently into Iraqi hands. When the Iraqi army was "rolled back" out of these countries into Iraq, he was willing to stop the war.

George Bush Jr. believed (or professed to believe) that the Iraqis were on the verge of obtaining weapons of mass destruction. To combat this "threat," he had to occupy Iraq to discover the existence (or lack thereof) such weapons.

The 1991 Iraqis had (some) modern weapons, but the soldiers were mostly poorly trained, supplied, and motivated. They were particularly lacking in transportation and communications equipment. That's why America's General Schwartzkopf opted for a "Hail Mary" attack around the flanks instead of a "head on" attack against which the Iraqis might have done some damage. In 2003, they were even more outmatched, except that they did have the "home court" advantage.

1. maybe they wanted to but political pressure from allies (especially Jordan, Egypt, and Saudi Arabia) to withdraw before reaching Baghdad made it impossible. In fact military leaders advised strongly to do just that, drive for Baghdad. Read general Schwartzkopf's autobiography (and that of president Bush Sr. for details)
2. arrest? On what grounds? Kuwait would have to press charges with the world court, and maybe did, but as the US doesn't recognise the world court it would be tricky for them to arrest anyone for delivery to the world court except maybe as a proxy for another country. When the US have apprehended foreigners abroad, it was always in international waters or airspace on charges for crimes in the US or against US citizens in international waters or airspace
3. Iraq's main failure was underestimating the international outrage and the speed and size of the response to their invasion of Kuwait. They still thought they were the darlings of the world for opposing Iran in the Iran-Iraq war (a war btw that Iraq started, not as often thought Iran). They never thought the world (and especially the middle east) would unite against them, or that Saudi Arabia would allow infidels to stage something like a hundred thousand troops inside its borders.

A quote from Dick Cheney, at that time Defense Secretary, circa 1991: "If you're going to go in and try to topple Saddam Hussein, you have to go to Baghdad. Once you've got Baghdad, it's not clear what you do with it. It's not clear what kind of government you would put in place of the one that's currently there now. Is it going to be a Shia regime, a Sunni regime or a Kurdish regime? Or one that tilts toward the Baathists, or one that tilts toward the Islamic fundamentalists? How much credibility is that government going to have if it's set up by the United States military when it's there? How long does the United States military have to stay to protect the people that sign on for that government, and what happens to it once we leave?"

Later, as head of Halliburton, Mr. Cheney had this comment: "[I]f Saddam wasn't there, his successor probably wouldn't be notably friendlier to the United States than he is. I also look at that part of the world as of vital interest to the United States; for the next hundred years it's going to be the world's supply of oil. We've got a lot of friends in the region. We're always going to have to be involved there. Maybe it's part of our national character, you know, we like to have these problems nice and neatly wrapped up, put a ribbon around it. You deploy a force, you win the war, and the problem goes away, and it doesn't work that way in the Middle East; it never has and isn't likely to in my lifetime."

Take a look at what happened after Saddam was disposed and you get the answer. Or take a look at what happened after the fall of Kadhafi.

Those dictators, although certainly not good people, but stabilized their regions. From a political point of view this is what is important. In the case of Iraq it was especially important for the US as a stable Iraq was a counterweight for Iran. Putting nice slogens about democracy, human rights, etc. aside, it was in the best interest of the US (and Western countries) to keep the balance and relative stability of that region intact.

So the question as I see is much more why did the US and Western countries dispose of Saddam and Kadhafi-

The causus belli of the First Gulf War was the Iraqi invasion of Kuwait. President George (HW) Bush managed to get a series of UN declarations calling for the Iraqi troops to leave, and authorizing succesively more drastic actions to achieve this, culminating in 678, which authorized "all nessecary means". This provided a certian amount of international cover for military action, but only towards restoring Kuwait, not toppling Iraq itself.

Also, many Muslim states signed on to the effort as allies, with the understanding that the recovery of Kuwaiti independence was the only goal. Most significantly, Saudi Arabia provided the required miltary basing and staging territory, with the understanding that the scope of the effort was limited to restoring Kuwait.

There is debate about whether the US administration would have liked to see Saddam toppled as well. Colin Powell claims in his memoirs that the intention was always to leave him in place, as a bulwark against Iran. However, various public statements made by President Bush led many people inside Iraq to (incorrectly) believe the US would support popular uprisings, with very unfortunate results. The administration of his son (GW Bush), certianly believed this was a mistake.

As to Iraqi failures, frankly they lost the war the instant it started. Their mid-20th Century vintage Soviet and Chinese military hardware was simply no match for modern top of the line US and NATO equipment. They could have been tactical geniuses throughout the conflict, and still gotten beaten handily.