The existence and length of the limitations on Presidential terms is something that is taken for granted now, but I think would be helpful to consider that term-limitation on the executive branch has been an ongoing debate in the United States since prior to its founding. Various plans during the Constitutional Convention offered plans ranging from no more than one term to life-service. At one point in the Convention, they actually approved a plan for a single 7 year term! Alexander Hamilton opposed and ultimately the open-ended four year term was what was implemented.

The New York, North Carolina, and Virginia ratifying conventions all called for amendments limiting terms to two. The rebuke was presented by Hamilton in the Federalist Papers, arguing that penultimate term (given term-limits) would be the last term that a president would have any "inducements to good behavior" given they would no longer face the need for approval by the electorate.

Flash forward 50 years and Andrew Jackson (in the wake of winning retribution from the 'Corrupt Bargain' of JQ Adam's 1824 election) called for a single 4 or 6 year term for the President in his first address to congress in 1829. In 1875 the Pennsylvania Republican convention passed a resolution of 'unalterable opposition' to Ulysses Grant running for a third term. Luckily, so did Grant and it did not come to pass. Nonetheless, congress in that year passed a resolution declaring that retirement after two terms was a 'time honored custom'.

Presidents Cleveland (of non-consecutive terms fame) and McKinley (of shot twice in the gut fame) both faced opposition from political bodies regarding third terms. McKinley solved his own problem by being assassinated, but it is worth noting that during Cleveland's second term that 13 amendment were proposed on the floor of congress which each sought to limit presidential terms.

McKinley's successor, Theodore Roosevelt, would placate the term-limit proponents as he regarded his first term as the three years spent completing McKinley's term. Roosevelt declared support for "the wise custom which limits the President to two terms... under no circumstances will I be a candidate for or accept another nomination", only to run for a third term in 1912. In response, the Democratic Platform was amended to call for a single 6 year term and Taft (Republican) would call on congress in 1913 to pass that limit. It passed the Senate but was not taken on in the house.

Woodrow Wilson also served two terms, but was too ill (after a debilitating stroke) to feasibly run for a third term.

In the midst of WWII Americans landed anywhere between 45% and 62% (according to Gallup) for an amendment limiting presidential terms. Whether FDR's attempts at court-packing and sweeping New Deal programs may have aided in that opinion, I don't know, but I think the idea that FDR's reelections were referendums on term-limits is unlikely since we were in a war at the time.

Ultimately, the US has spent more that 'a scant 7 years' considering limiting terms - rather its been lurking under the surface of the country's politic for it's entire lifetime. That term limits were placed after FDR and not another president is partly due to circumstance, assassinations, or war. I think FDR's four terms were perhaps what pushed the amendment into passage, but it wasn't a sea change or a spite against FDR, necessarily.

Source: The Twice and Future President: Constitutional Interstices and the Twenty Second Amendment - Bruce G Peabody and Scott E Gant.

It's complicated, and it's fair to say there was more than a bit of political partisanship going on.

The call for a presidential term limit began even before FDR was elected to his third term in 1940, so it should be thought of as an argument of 12 years rather than seven.

"The dangers of continuing any president in office more than two terms are no less real because the policies he advocates may be sound," said Columbia University professor Dean Smith to a U.S. Senate group in 1940.

Smith, and a large number of Congressional Republicans viewed FDR's pursuit of a third (and fourth) term as dangerous because it broke the two-term precedent set by George Washington and opened the door for future presidents to do the same. Particularly after the war began, Congressmen were careful to avoid criticizing FDR ─ who was enormously popular ─ but they said that future presidents might not be as skilled or tempered, and there was a danger of totalitarian rule in the United States unless limits were set on their terms.

The U.S. House's vote in 1947 was 285-121, with all 238 Republicans in favor and 47 Democrats supportive as well. That gave it a margin of 14 votes beyond what was needed. Opponents said it would "shackle future generations" who might need a president to stay in office during an emergency, just as FDR did. But that argument was defeated by those who said (as Sen. Alexander Wiley of Wisconsin did) that "too long occupancy of the Presidential office always makes for danger of dictatorship."

In the Senate, Democrats garnered enough support to amend the proposed Amendment so that Truman ─ who ascended to office not long after FDR was re-elected ─ could serve longer. That was subsequently changed so a president can only serve a maximum of 10 years.

During the Senate's vote, Sen. Claude Pepper, D-Florida, said the Amendment was "aimed directly at Franklin D. Roosevelt" and if it had been enacted in 1940, it would have "thwarted the will of an overwhelming majority of the people."

So the Roosevelt argument was raised in both the House and Senate, but it didn't stop the proposal from advancing.

There was a pretty lengthy argument against the Amendment's approval published by the Times shortly after the Senate voted to approve it. You should check it out if you're interested in the arguments against the Amendment.

The arguments for the Amendment were much simpler: It was a step against dictatorship and tyranny, and it made law an implicit precedent set by the nation's founders. With Republican legislators controlling statehouses and both houses of Congress for the first time since Herbert Hoover, the proposed Amendment had ample support. By 1951, there was even competition among states, which raced to see which would become the 36th and clinching state.

Adding to the question, if not spite against FDR, was it in honor of him?