Legend has it that George Washington was offered the title of King of America in 1783 and thought that the idea was abhorrent, however there doesn't seem to be any evidence for this. We do know that George Washington didn't pass into a third term of the Presidency but that's more because that by that stage he was 65 and probably a very tired man. His retirement only lasted two years and closed with his death just 33 months into his retirement.
But I wonder what would have happened had he been made king.
George married Martha Dandridge Custis in 1759 but because George had fought off a case of smallpox in 1751, they were unable to have any children. Most of George's estate passed to his nephew Bushrod Washington, whom he was very close to.
Admittedly the line of succession appears very muddy but thankfully someone at Ancestry.com has already looked into this and we have a sort of incomplete line:
House Of Washington
George - 1776 - 1799
William I - 1799 - 1810
Bushrod I - 1810 - 1831
Spotswood I - 1831 - 1865
Bushrod II - 1865 - 1918
James - 1918 - 1924
Spotswood II - 1924 - 1948
William II - 1948 - 1994
Paul - 1997 - date
Presumably the United States would have probably adopted a parliamentary democracy mirroring that of the United Kingdom. Maybe the US House of Lords would have been elected, maybe not, though it's fair to assume that the executive of the nation would have been formed in the US House of Commons and not directly appointed by the head of state as is the case now.
It is a real irony that in spite of a deep-seated fear of kings and of papacy; one only needs to read Thomas Paine's 1776 treatise "Common Sense" (and I think extremely poorly researched) to see this; that the President of the United States in terms of the exercise, has wielded far more actual power than any monarch from George III onwards ever has. The Declaration of Independence itself is a constant attack on the person of George III, even though he probably gave nary a thought to America at all.
In the capacity of Commander-In-Chief of the armed forces, the President of the United States has declared war 10 times, been authorised by Congress to make war a further 12 times, been funded by Congress to enable military engagements 7 times and has acted without any express authorization from Congress on at least 125 times. In the United Kingdom, even though the Monarch has the power to declare War, not even once since 1776 have they done so; with it all being done by the parliament or the Prime Minister.
Maybe if the executive of the nation and specifically the power to commit the nation to war had been held by the parliament, then maybe the character of the nation would have been different.
The logical comparison of what the Kingdom of the United States of America would have looked like is either Canada or indeed Australia. Australia elects its Upper House whereas Canada does not and Australia does employ a system whereby the seats try to give an equal say to the states.
Bills of Supply, that is monetary bills and the budget, would still pass through the parliament in the normal fashion, though I suspect that unlike now, the tendency for filibustering would not show itself quite so much.
Obviously where one stands depends on where one sits and the animosity for having a king was bitter following the events of 1775–1783. I can't help but feel though that the United States as a nation would have projected a softer and dare I say it, kinder, shadow into the 19th Century. We shall never know though.