I don't ask this just to be speculative. The reason I ask is because many indicators of authoritarian overthrow were present in his case, and he acted like a sort of North American caudillo. He was a famous and popular war hero general with military support. He ran as a populist, and was really the only major 'people's candidate' in US history up to that point. Arguably the first true political party in US history was built around him, and it was well organized and powerful for several years before the Whigs organized in opposition. After the election of 1824 and the 'corrupt bargain' between John Quincy Adams and Henry Clay put JQA in the presidency, he had a legitimate political grievance, having won the plurality of both the popular vote and the electoral college. Yet he waited four years to run again. He ignored the rule of law in major ways, both as general in his invasion of Florida and as president when he ignored a Supreme Court decision. Even then, the strongest move congress made was to censure him. He brought in a spoils system for federal jobs, which essentially meant anyone who could be hired to a federal position had to play by the rules of his administration and party. Many of his critics at the time accused him of behaving like a king. In many other young constitutional democracies, a general such as himself would have staged a coup after the corrupt bargain, or after becoming president would have kept running in sham elections every four years claiming the country needed him, ignoring checks on his power and continuing to erode constitutionality. So why didn't he do this? Did he believe in the system overall and truly not want to overthrow it? Did he simply not want the job indefinitely? Would the rest of the Democratic Party have gone against him? Would the people actually realize what he was doing if he stayed in office and then rise up against him? Was free press an effective check?
You're right that "could have" dips into speculative history, since we know that Jackson didn't become a king. Historically and politically, it wouldn't have been in his nature to do that because one of the planks in his platform was term limits.
Part of the reason why this came up was in response to the Tenure of Office Act passed by Congress in 1820. The act was very much in the public debate both before and during Jackson's presidency. The act limited terms of public service, but officeholders could be renominated and reappointed indefinitely to the same job they already had, which seemed to encourage appointees to do corrupt things to please their appointers. Jackson and his supporters in Congress attempted to reform this law. Their efforts were ultimately unsuccessful, and the law remained in place until 1867.
Seeing as Jackson was a proponent of this reform, for him to try to hold onto his own public office despite his efforts to get everyone else to give up theirs wouldn't necessarily have played very well with his base.
These ideas were related in his First Annual Message to Congress, delivered on December 8, 1829. Here's a lengthy quote (emphasis mine) to get an idea of Jackson's thoughts on the subject of unlimited terms in office:
There are, perhaps, few men who can for any great length of time enjoy office and power without being more or less under the influence of feelings unfavorable to the faithful discharge of their public duties. Their integrity may be proof against improper considerations immediately addressed to themselves, but they are apt to acquire a habit of looking with indifference upon the public interests and of tolerating conduct from which an unpracticed man would revolt. Office is considered as a species of property, and government rather as a means of promoting individual interests than as an instrument created solely for the service of the people. Corruption in some and in others a perversion of correct feelings and principles divert government from its legitimate ends and make it an engine for the support of the few at the expense of the many. The duties of all public officers are, or at least admit of being made, so plain and simple that men of intelligence may readily qualify themselves for their performance; and I can not but believe that more is lost by the long continuance of men in office than is generally to be gained by their experience. I submit, therefore, to your consideration whether the efficiency of the Government would not be promoted and official industry and integrity better secured by a general extension of the law which limits appointments to four years.
In a country where offices are created solely for the benefit of the people no one man has any more intrinsic right to official station than another. Offices were not established to give support to particular men at the public expense. No individual wrong is, therefore, done by removal, since neither appointment to nor continuance in office is a matter of right. The incumbent became an officer with a view to public benefits, and when these require his removal they are not to be sacrificed to private interests. It is the people, and they alone, who have a right to complain when a bad officer is substituted for a good one. He who is removed has the same means of obtaining a living that are enjoyed by the millions who never held office. The proposed limitation would destroy the idea of property now so generally connected with official station, and although individual distress may be some times produced, it would, by promoting that rotation which constitutes a leading principle in the republican creed, give healthful action to the system.
The Office of Tenure Act wasn't the only reason Jackson was a proponent of term limits. If you read the rest of Jackson's full First Annual Address to Congress, it's evident that his thoughts on the subject were very much influenced by the election of 1824, which he lost. He had won the popular vote, but not enough electoral votes to win the Electoral College, so the election got thrown to the House of Representatives. There, he lost by a single vote and John Quincy Adams became president. This incensed Jackson because A) Adams was a member of the House, so he got to vote with Massachusetts' electorate, and B) Adams then appointed a lot of the House members who had voted for him to positions of public office. Jackson felt this had been allowed to happen because the members of Congress were no longer answerable to the public but were doing what was in their own personal best interest.
To address this issue, Jackson went on to propose in his address a Constitutional Amendment to abolish the Electoral Congress in favor of a direct popular vote, and also:
In connection with such an amendment it would seem advisable to limit the service of the Chief Magistrate [i.e., the President] to a single term of either 4 or 6 years.
Of course, when that Amendment never happened, Jackson went ahead and served for a full eight years. That said, he called his second term "unsolicited" in his Second Inaugural Address, and he kept the line in about the 4 or 6-year term limit on the Presidency in the first six of his Annual Addresses to Congress, and called attention to Presidential election reform in his last one as well.
The proposed amendment was introduced in Congress 21 times but never passed both houses, and most times it was proposed, it didn't even get a vote.
Jackson's "do as I say, not as I do" stance on this issue did cause some complaint, even among members of the Democratic Party. When Martin Van Buren became Andrew Jackson's hand-picked successor to run in the 1836 election, Jackson's own former Secretary of the Treasury William Duane wrote in a letter to a friend:
"It is the fact that men change that makes a republic preferable to a monarchy. Washington and Jefferson would never trust themselves with power longer than eight years. General Jackson was, at one time, so fearful of the influence of power and passion upon himself, that he was in favor of limiting service in the presidency to four years. But what a revolution we behold! Now he is not only content to retain power for eight years, but desirous to transfer it to a favorite! Such is the effect of power and flattery! Are you amazed? I am not..."
One biographer stated that there was "some talk" about running Jackson for a third term, but his health was failing by 1836, and precedent was already set against it. Further, in the words of political rival John Quincy Adams, by 1836, Jackson had "wearied out the sordid subserviency of his supporters." There was always the risk that Jackson would lose a third term, either at the Democratic nominating convention or in the general election, and tarnish his legacy.
Another political opponent, John C. Calhoun, wrote privately that he'd heard rumors that Jackson would be open to the idea of a third term "only" if war broke out with France, and Calhoun suggested Jackson was then planning on starting a war with France in order to hold onto power. Of course, this didn't pan out, and the rumor that Jackson would possibly accept a third term may have been just that: a rumor.
Maybe Jackson could have run again and been elected to a third term? But he didn't so we can never know. Given that term limits for the President and Vice President were part of his platform since he took office in 1828, it would have raised a few eyebrows even among his own supporters for him to so flagrantly break with himself and with tradition to hold on to office longer than eight years. Ultimately, Jackson never publicly suggested running for a third term, and everything he ever said or wrote was on the side of limiting the President's term in office.
FURTHER READING:
The History of the Democratic Party from Thomas Jefferson to Grover Cleveland by Chandos Fulton, 1892
"Third Term Issue Ranks High In U.S. Political History" by Joseph Reed, Washington Post, March 17, 1940
Third Term for President of the United States: Hearings Before a Subcommittee of the Committee on the Judiciary by the U.S. Senate, September-October 1940