The opinion has 64 pages and the dissents another 90, so it will take me a while to read through it, and I'll comment on it as I go. Here is the first instance of the typical Scalia sacrifice of logic for rhetoric (pp. 6-7):
What is more, in all six other provisions of the Constitution that mention "the people," the term unambiguously refers to all members of the political community, not an unspecified subset. (...) This contrasts markedly with the phrase "the militia" in the prefatory clause. As we will describe below, the "militia" in colonial America consisted of a subset of "the people"---those who were male, able bodied, and within a certain age range. Reading the Second Amendment as protecting only the right to "keep and bear arms" in an organized militia therefore fits poorly with the operative clause's description of the holder of that right as "the people."
There are two logical problems with this statement. First, Scalia is ignoring plausible alternative readings that avoid this purported tension, and second, his insistence on "a subset" is tenuous.
If the Second Amendment is read to guarantee a right to "keep and bear arms" for militia-related purposes, and that right belongs to "the people", then it also implies the right of "the people" to serve in the militia. In other words, it prevents a permanent segmentation of the society into an armed military class and a disarmed one. That couldn't have been a trivial issue in 18th century, so this is at least a plausible reading of the amendment, and one which uses the terms "the people" and "militia" in a fully compatible way.
Also, being "male, able bodied, and within a certain age range" is hardly a restrictive set of conditions. The political community of the time included only men, and of course only adult men, so exclusion of women or underage men is already implied in "the people" as Scalia defines the term. We thus only need to consider the significance of the exclusion of old and disabled men for the distinction between "the people" and "militia" as its subset. But a musket was not of much use to many a disabled man (particularly if the disability had to do with upper extremities or teeth), and an older but healthy man, although not required to serve in the militia, wasn't necessarily forbidden from doing so. Therefore, Scalia's distinction is meaningless.
And yes, he tries to justify it in the next several pages, but his arguments just get more and more strained, culminating with a false analogy on p. 13, where he writes that (because of how he parses the idiom "to bear arms") using the phrase "to keep and bear arms" to imply a military context would be like saying "he filled and kicked the bucket" to mean "he filled the bucket and died". He calls that grotesque, which it is, but mainly because of his own mangling of the relations between words.