The most thorough examination of this issue is found in Bruce G. Peabody's article "George Washington, Presidential Term Limits, and the Problem of Reluctant Political Leadership", published in Presidential Studies Quarterly in September 2001.

As Peabody argues, Washington wasn't motivated to retire by a belief in president term limits. In 1788, a year before taking office, he wrote a letter to the Marquis de Lafayette saying he didn't believe a President should become ineligible "after he should have served a certain course of years":

"Guarded so effectually as the proposed Constitution is, in respect to the prevention of bribery and undue influence in the choice of President: I confess, I differ widely myself from Mr. Jefferson and you, as to the necessity or expediency of rotation in that appointment. The matter was fairly discussed in the Convention, & to my full convictions...There cannot, in my Judgment, be the least danger that the President will by any practicable intrigue ever be able to continue himself one moment in office, much less perpetuate himself in it—but in the last stage of corrupted morals and political depravity: and even then there is as much danger that any other species of domination would prevail. Though, when a people shall have become incapable of governing themselves and fit for a master, it is of little consequence from what quarter he comes...I can see no propriety in precluding ourselves from the services of any man, who on some great emergency, shall be deemed, universally, most capable of serving the Public."

In modern terms, he believed the Constitution contained enough safeguards to keep immoral or politically corrupt individuals from continuing in the office of President, and if a President somehow continued in office despite those safeguards, it meant the people have "become incapable of governing themselves", and the country has worse problems going on than term limits. Term limits could just lead to another branch or office of government attaining out-sized power, and the country is better off being open to a long-serving President in case of "some great emergency" that the President would be the "most capable" of handling.

Citing several earlier historians, Peabody argues that the general academic consensus is that Washington almost certainly would have been elected to a third term if he'd wanted it. In fact, there were calls for him to continue serving. In 1796, a friend of Washington's nephew George Lewis wrote a letter to Lewis beseeching the nephew to try to talk Washington into serving a third term, because "many thousands" of Americans would mourn the day he left office: "[I]f our worthy friend give us up, it will through a damp upon the spirits of all good friends to Government" and will embolden the "Enemies to our blessed Country". But if Lewis did actually lobby his uncle, Washington turned down the proposition.

One election cycle later, there was another effort to draft him. In 1799, the Governor of Connecticut, Jonathan Trumbull Jr., wrote to Washington to persuade him to reconsider his retirement and run for a third term. Trumbull was a friend of Washington's who had been a Federalist and Congressional ally during Washington's presidency. John Adams' presidency was seen as being less than successful by that point, and he was likely to lose re-election. Trumbull wrote:

"Another Election of a President is near at hand, and I have confidence in believing, that, should your Name again be [brought] up, with a View to that Object, you will not disappoint the hopes & Desires of the Wise & Good in every State, by refusing to come forward once more to the relief & support of your injured Country. Need I apologize to you Sir! for this hint? Or shall I frankly tell you, that this Idea is not vaguely started by me, but is strongly prompted by the necessity of our situation, and may probably be pursued in earnest; for unless some eminently prominent Character shall be [brought] up to view on the Occasion, the next Election of President, I fear, will have a very ill[-]fated Issue."

Washington turned down the suggestion, writing that he would be "charged not only with irresolution, but with concealed ambition". He preferred "to pass through the vale of life in retirem[en]t, undisturbed in the remnant of the days I have to sojourn here".

If not motivated by term limits, then, what were Washington's reasons for his initial retirement after his second term in office? Peabody writes:

"Washington's decision to leave office seems to have been inspired by three intertwined factors: a desire to strengthen the institution of the presidency, his longstanding interest in securing personal honor and reputation, and a deepening weariness with public service (and a corresponding wish to return to his private affairs in Virginia)."

Peabody argues that Washington was aware of a "pervasive domestic distrust of federal executive power" and his retirement was an effort to assuage these fears. One of the criticisms of the federal system, and of the Constitution, was that it gave too much unchecked power to the executive branch. The Anti-Federalist Paper "Cato No. IV" (written by New York Governor George Clinton) argued that the President's lack of term limits would prove to be detrimental to a republican form a government. And despite Cabinet positions requiring Senate confirmation, the President would "generally be directed by minions and favorites" and not by elected officials, or else "a council of state will grow out of the principal officers of the great departments [i.e., the Cabinet], the most dangerous council in a free country."

Philadelphensis (Benjamin Workman) went even further in his Anti-Federalist No. 74:

"The President-general, who is to be our king after this government is established, is vested with powers exceeding those of the most despotic monarch we know of in modern times. What a handsome return have these men [the authors of the Constitution] made to the people of America for their confidence! Through the misconduct of these bold conspirators we have lost the most glorious opportunity that any country ever had to establish a free system of government. America under one purely democratical, would be rendered the happiest and most powerful nation in the universe. But under the proposed one composed of an elective king and a standing army, officered by his sycophants, the starvelings of the Cincinnati, and an aristocratical Congress of the well-born -- an iota of happiness, freedom, or national strength cannot exist."

With his retirement, Washington aimed to prove that these fears by Anti-Federalist critics had been unfounded.

On Peabody's second point, the author argues Washington had a history of being concerned with securing his social standing, which, in late 18th Century Virginia society, meant "the public exhibition of distinguished service and virtuous political behavior, especially disinterested commitment to the public good" in the eyes of his upper-class Virginian (and American) peers. To that end, when Washington turned down Trumbull's plea to re-enter politics, he wrote of his fear of being received as an old, weak imbecile.

Author Garry Wills examined this point at length in his book Cincinnatus: George Washington and the Enlightenment. Washington had been a founding member of the Society of Cincinnati, named for the Roman leader Cincinnatus who had assumed absolute power in Rome during a time of war, only to relinquish the power when the war was won, and went back to being a civilian landowner. By all accounts, Washington believed in this ideal, and wanted to be applauded by his contemporaries for being a similar public figure.

On Peabody's third point, the author argues Washington was very much motivated by a desire to retire, fed up with politics and public life. He had not been particularly enthusiastic to serve as President in the first place. He had first indicated he was going to retire after a single term, and it was only with the intervention of Alexander Hamilton, Thomas Jefferson, and James Madison that he reconsidered. The increasing partisanship of the era also convinced him to stick around for a second term, and spoke out about the situation in his Farewell Address. But his distaste of politics by the end of his second term outweighed his concern that the partisanship threatened the future of the republic. Nonetheless, it did worry him, and he promoted the idea of a united "national character" over sectionalism and political infighting so that the republican form of government under the U.S. Constitution would have lasting power.

TL;DR: While his retirement may not have been motivated by a belief in term limits, it surely was motivated by a belief that public officers should be willing to relinquish power as a republican ideal. And this ideal, he believed, would strengthen the United States' republican form of government both in the short term and the long term. It had the added bonus of burnishing his own public image. Between his service as the head of the Continental Army and then of the executive branch of the U.S. Government, he'd had enough of public life by that point, and was glad to walk away, and enjoy a much-deserved retirement.

Thomas Jefferson and James Madison, on the other hand, more truly believed in term limits for Presidents as an ideal of its own, and the effort to institutionalize them was a long-fought, if intermittent, battle, culminating in the 22nd Amendment to the U.S. Constitution.