They might have, we may never know if your hypothetical is true or not because Obama cut and run before we had a chance to see, but regardless Obama was not forced to do anything even if people did not like that a certain thing happened. We have never been a direct democracy, we are a representative republic.
Not renewing the agreement was pretty much Iraq's way of saying we don't want you here any more, so you would have been staying against the will of the government. This is not a great long term strategy.
But let's say Obama decided to 'tough it out'. What next?
When Obama did normal stuff, many on the right saw him as a subversive traitor trying to sabotage the US. It doesn't take a rocket scientist to work out how they would respond to Obama allowing a Muslim country to arrest and kill US soldiers with impunity.
It would most likely make any person who supported it unelectable, so would face widespread opposition in both houses. I'm not sure what actions they could take to force Obama's hand, but they would certainly try.
That is patently false, whatever event real or imaginable you mentioned above what you describe actually happened in thousands of cases in WW2. Did Truman or Roosevelt tuck tail and run? Nope, Truman dropped the most destructive weapon ever used in the history of warfare on the Japanese and they ground Hitler's Germany to dust. Did Lincoln fold in the wake of draft riots, tens of thousands of northern soldiers being beaten and starved to death in prison camps, and virtually every one in the government at one time or another saying he should give it up? Nope, he stuck it out to the bitter end and is praised as perhaps the greatest president we ever had. I can keep showing you your points are meaningless and that you (as those who usually defend failed liberal policies) do not know anything about history, but it won't help.
First of all, it wasn't a 'liberal policy', Bush decided to "cut and run" when he signed off on a withdrawal by 2011. Also neo-conservative/liberal interventionist nation building spans both left and right, as does opposition to it.
If you want to look at history then it's just as easy to find examples of people who 'stuck it out' with disasterous consequences. The Persian desire to conquer Greece end up with the Greeks conquering Persia. Napoleon toughing it out in Russia didn't end to well for him either.
Also, this was no WW2, it was an unpopular war with an unrealistic objective of nation building.
Unless the objective was an unending game of whack a mole against insurgents with no end game in the face of increasing hostility and resentment, what do you think could have been achieved?
The Iraqi Army was not going to be ready any time. As you saw when they fled from a handful of jihadis, they were simply a cash cow for politicians and officers who were kept deliberately weak. A powerful military would be a threat to the politicians.
Were you born 20 years or less ago or something, this kind of stuff is not new, it is not unique, and it has occurred in massive quantities in our own past and in the past for every nation engaged in a significant war. What is new is the whining and crying the moment anything gets hard, and our having so petty and weak a leader the he cannot stay the course in the face of adversity. Why don't you go back and ask general Washington and his frozen men at Valley Forge who were facing the greatest military power in history at the time if a few people being jailed or killed was too high a price to pay?
The political reality of the 21st century is very different. No politician could survive such a humiliation. It wouldn't be seen as 'toughing it out', but the most abject surrender and weakness of any president in history.
You would literally be paying billions of $$$ to a government that was humiliating you in public in conjunction with Iran.
Obama would be the most hated President in history and a one term lame duck rejected by even his own party.
When we went into Iraq the second time (BTW I was in the military for both wars so I had a bit on an inside take on the events) no one thought that the insurgency at the close of what was about as costless and easy a victory as the most optimistic could even dream of, would be as massive and sustained as it was. It was purported to be this horrific thing that was getting out of hand and going to overwhelm us. However the fact of the matter is it was one of the best gifts we could have wished for. The worst part of fighting terrorists is that they hide and blend in, but in Iraq they were actually hunting us by the thousands. All we had to do was dig in and use our technology to get kill ratios of dozens or hundreds to 1. This was reported to be a nightmare but if we actually were at war with terrorism you could not have wished for better. So it depends on the goal which makes a negative or a positive.
I have some questions that I'd be interested in your perspective on. As someone who served in the military, how would you feel if service men and women, doing a tough and dangerous job, also had to do this under the knowledge that at any time they wanted the Iraqi police could arrest them on a whim? How do you think morale would be? How would you feel if the police marched into your base and took your friends away to probably be tortured and possibly executed? How would you feel the 2nd, 3rd and 4th times after seeing people like yourself paraded on TV and publicly humiliated? Do you think heavily armed people would just stand by and do nothing? Would you honestly still be praising Obama for showing toughness and moral fortitude?
The Iraqi government is significantly influenced by Iran. Don't you think that Iran would have taken the first opportunity to pick itself up a load of bargaining chips in the form of US military personnel?
Not just the front line soldiers either, the entire command structure would be subject to arbitrary detention, easily justified by any civilian casualties. I'm not sure how the military could function under such conditions.
The other alternative would be to reject Iraqi sovereignty and stay there by force, but I don't really see how this would be helpful if the goal was creating long term stability.