We might well have let him do just that. As one of the only universally popular founding fathers, George Washington was considered a true American hero at the time. Alexander Hamilton was not averse to the idea of having our own King George as opposed to the British version.
Most likely, Washington would never have accepted such a position, if for no other reason than he was tired, having served his country in almost every manner he could since before the Revolution. If he had become King, it's also quite possible he would have been just as wise, just as considerate of individual rights as he was in his Presidency. That's what we call a "benevolent monarch". Luckily, he was also wise enough never to let us become a monarchy at all.
George Washington probably would have made a pretty good king. He was tall and dignified, he was a great warrior, and he commanded the respect of most of his countrymen. He was also humble enough that he would have taken good advice from some of his talented courtiers, people like Alexander Hamilton, John Adams, Benjamin Franklin, etc., etc.
One major problem is that he had no children. Who would take over after his death? There was no one so universally popular as Washington who could be appointed king. So, one of two things might have happened:
A peaceful election for King (or President) might have been held; or, there would have been a bloody feud between some of the men mentioned above.
Although one never knows, I think we should thank Washington for refusing to be King.
At the very least, the history of the United States would have been very different.
It is not a direct analogy, but I think that the experience of Mexico might be helpful.
When they became independent, Augustin Iturbide allowed himself to be made emperor. From then on, instead of having elections, they pretty much had a series of civil wars between monarchists and people who wanted a republic.
So I can imagine Jefferson and his people leading a rebellion and Hamilton and his people fighting the Jeffersonians... Not good.
Many "historians" have put forth the idea that Washington could have been king... I believe that he could not have, even if he had wanted to. First of all he had never indicated the slightest desire to actually become king... nor was it EVER offered to him... nor did ANY of his advisors ever advise him to try to become king. Over time some historians have put forth the idea that of all the founders Washington might have been the only one with the outside chance to do it ... but I believe that it is all simply admiration in hindsight.
If Washington was to become King...he'd have had to "take it" ... it would never have been offered to him. Considering the previous war to gain independence from a KING... there would have been a huge backlash from all sides of the population... Washington's legacy would have been ruined....and the country would have disolved in a mess of civil war and immediate foreign nations attempting to re-gain the country during the massive unrest.
Also at the time of Washington's first election...he was not old and/or tired. He actually presided over the Constitutional convention itself and accepted the office ready and willing to lead ... and he served 8 long years only to die of pnumonia at 67.
It was not just up to George Washington to decide to become a president or king. There were many prominent leaders that were involved in American war of independence. Further their actions were guided by a common agreed ideology and plan of actions. Part of this ideology evolved as the Revolution progressed and was partly expressed in documents like Declaration of Independence (July 4, 1776), Articles of Confederation (1781), and American Constitution (1787).
Also it was not as if Washington decided to become even president. He was elected to become a president, under the constitution of the country.
It is a a highly hypothetical question, which can have no definite answers. Different people can have different guesses about what could have happened. I believe if Washington had tried to become a King of the United States of America, or even of one of the states within it, only one or both of the following two things could have happened. First possibility is that it would have adversely impacted the course of American War of Independence. Second is that George Washington would not have achieved an eminent position among the founding fathers of USA. Perhaps he would have been discarded by others important leaders as soon as his intentions were known to them.