Writing for the five-justice majority in District of Columbia v. Heller, the 2008 decision in which the Supreme Court recognized that the Second Amendment protects an individual right to armed self-defense, Scalia said that right was violated by a law that banned handguns and required owners of long guns to disable them with trigger locks or keep them unloaded and disassembled. But he strongly implied that nearly every other existing gun control law would pass constitutional muster.
Scalia wrote that "nothing in our opinion should be taken to cast doubt on longstanding prohibitions on the possession of firearms by felons and the mentally ill, or laws forbidding the carrying of firearms in sensitive places such as schools and government buildings, or laws imposing conditions and qualifications on the commercial sale of arms." He also said the Second Amendment applies only to weapons "in common use for lawful purposes," a condition that seems designed to justify any pre-existing national ban, and mentioned "prohibiting the carrying of 'dangerous and unusual weapons'" as "another important limitation on the right to keep and carry arms."
Heller thus blessed a policy that permanently disarms people who have never committed a violent crime or shown any propensity to do so and left open the question of whether and where people have a constitutional right to carry guns outside their homes. It gave wide latitude to "conditions and qualifications" for selling guns, which might including licensing and registration of buyers as well as background checks and waiting periods and provided a potential rationale for outlawing guns that politicians deem especially dangerous (such as "assault weapons"). In another 5-to-4 decision two years later, the Supreme Court overturned Chicago's handgun ban, confirming that the Second Amendment applies to cities and states as well as federal domains, but leaving its scope unclear.
Defining the scope of the Second Amendment is where the legal action is now, and that could remain true even if a Democrat picks Scalia's successor. Rather than overturn Heller completely, a new Supreme Court majority friendly to gun control laws could simply read the right to arms narrowly enough to accommodate them.
There were two dissents in Heller, both joined by all four justices in the minority. One dissent, written by John Paul Stevens, argued that the Second Amendment does not protect an individual right to armed self-defense. The other dissent, written by Stephen Breyer, argued that the D.C. handgun ban was consistent with that right.
By embracing Breyer's reasoning, which tolerates wide variation in gun controls based on local conditions, a new majority of Democratic appointees could approve whatever restrictions come their way while still claiming to respect Second Amendment rights. That is essentially the position taken by Barack Obama and the two Democrats vying to replace him, Hillary Clinton and Bernie Sanders.
Politics would once again be the main constraint on gun laws, meaning that jurisdictions like D.C. and Chicago would once again be free to ban handguns. In other words, the most likely short-term consequence of adding another Democratic nominee to the Supreme Court is that Democrats would succeed in disarming their own constituents.
At the national level, existing laws would be upheld, but that is probably also true under Scalia's approach. New gun controls -- whether the universal background checks Democrats are pushing or the Australian-style mass confiscations that Clinton admires -- would not suddenly be more likely to pass.
That could change over the long term, of course, which is why constitutional constraints are important. You don't wait for a fire to break out before you buy an extinguisher.