Scalia's opinion is that, effectively, the courts should be judging based on Scalia's personal opinion of the "legal culture’s conception of constitutional interpretation". He thinks the culture no longer interprets the constitution in, well, "the right way".
In the previous paragraph he points out how, effectively, any majority-leaning set of judges can just invent new rights to assign to the country and they will be permanent [or at least, until a later court overturns them]. And he's right. But this is not news. Presidents have been trying to assign judges that lean their way on issues since... a long-ass time. The judges are human, moral, and they change their minds about how to interpret the law as the mob's mood changes.
And he's right that the majority simply shoving it's way down the minority's throat is unjust and is going to breed contempt. When the North defeated the South, racial tension didn't just disappear overnight. Those in the South kept a grudge and an ignorance that persists to this day. We may be using the power of the majority to force the removal of the flag, but those people who find that force offensive will not forget it. We didn't convert them, convince them, or make them understand; we held the proverbial Public Relations gun to our leadership's heads.
You ask why the states shouldn't have resolved the issue over time. Should the federal government have allowed some states to continue slavery? We all know it's a crime against humanity now, but what actually makes it a crime? Is it some old english scribbled on some parchment centuries ago? Or is it merely an intangible, unwritten law that reversed itself naturally as society changed over time?
The truth is, human civilizations have been morphing and evolving for thousands of years and they aren't going to stop doing so because some old farts had an idea a couple hundred years ago and we're trying to stick to their idea as hard as we can. Never changing, no matter how hard our society demands it, becomes simple fundamentalism, and our country was not founded on fundamentalism. Our country was founded - no matter how trite this sounds - on the idea of liberty over tyranny. We're always going to lean more towards liberty than tyranny.
We have a pretty good track record so far of not letting the states or federal government erode our liberty just because of the color of our skin, or our gender, or sexual identity, or religion. We protect people, always. But we don't limit people just because we don't like them or are afraid of them. Any time a state is asking to do that, we will eventually push back hard enough that the state will lose.