Is HBO's Generation Kill a musical?

No. Musical films typically use full-scale song and dance routines in significant ways, such as to advance the plot. The use of music in Generation kill is much much more subtle, usually giving some more insight on the marines' personalities or situation. Basically, they are singing to alleviate boredom when they are driving to the next objective rather than breaking into full song and dance routines ala the gang fights in West Side Story. In fact, during the audio commentary, one of the creators mentions that he sees the series as a sort of long road movie.

Questions of the form “does x belong to category X?” are either meaningless or trivial, and the answer depends only on definition of X. Depending on how you define the word “musical”, Generation Kill can either be a musical, or not. That's it, there's nothing else to say.

Wiktionary describes musical as “a stage performance, show or film which involves singing, dancing and musical numbers performed by the cast as well as acting”. Generation Kill involves singing, so according to this dictionary definition it's a musical, sorta.

Wikipedia page on musical film describes it as “a film genre in which songs sung by the characters are interwoven into the narrative, sometimes accompanied by dancing” and also clarifies that “[t]he songs usually advance the plot or develop the film's characters, though in some cases they serve merely as breaks in the storyline”. Generation Kill uses songs as breaks in storyline, but not to advance plot. So it can or can't be a musical, depending on how many factors you require for it to qualify as a musical. Bollywood films feature songs and dancing, are they all musicals?

Clearly, arguments from dictionary are useless, because dictionaries are for documenting common usage of words, they do not prescribe how you should use the word. Words don't have default, “correct” meanings, it's us, humans, who assign meaning to every word for convenience. Words are just tools that we use to efficiently exchange information.

It is true though, we should use words efficiently, not to confuse people. Words are used best, when they describe a dense similarity cluster, a group of things which look very-very similar to each other. How similar is Generation Kill to My Fair Lady, Rocky Horror Picture Show, Chicago? If the last 3 are pigeon, flamingo, and sandgrouse, Generation Kill is more of a penguin.

So let's not try to pigeonhole the show into different arbitrary categories, because it's like asking “But is Pluto really a planet or not?”. There's a real question you want to ask. You want to gain some insights from the way songs are used in the show, right?

In addition to Evan Wright, who wrote the original book, the show is written by David Simon and Ed Burns, who created critically acclaimed The Wire. The Wire is known for it's gritty realism, density of plot and characters, even comic relief scenes are there to make a point. The Wire is deliberately journalistic in its style, owing to David Simon's background, it looks like a CCTV footage. Except opening and closing themes and songs at the end of closing episodes of each season, The Wire's music is diegetic, which organically fits the style.

Filmmakers that are trying to make a point, or convey some idea, or document life as it is in an art form, frequently consider non-diegetic music (and some other special effects and modifications) as a cheap way to increase dramatism or entertainment factor. Sometimes they consider such modifications and filmmaking decisions a trick to make viewers stick to screen by exploiting their cognitive biases or desire for cheap thrills, akin to laugh tracks. They think art should mesmerize and thought-provoke on its own, via plot, actors performances and artistic value, not by using gimmicks.

For example, Dardenne brothers' film L'Enfant has no music, except once when heard from the car radio.

A sound editor of The Wire described Simon's approach to non-diegetic soundtrack in a Reddit AMA:

David Simon is very opposed to many of the ways that typical TV audiences expect to be cued or manipulated into thinking or feeling a reaction to a scene. He's the guy who, if he did write a sit-com, would not allow a studio audience or laugh track. He views musical score the same way. One of my crew's challenges, then, was to find ways to evoke mood with backgrounds.

Simon carried this attitude to Generation Kill. It's not an action film, it's not for cheap laughs or thrills, so why would there be a need for background music, that is not part of the scene? On the other hand, how would a bored, expressly indifferent, machistic soldier entertain himself on a road under a hot desert sun? Sing cheesy pop songs of that period. Here, you have both a soundtrack and a character behavior that is completely authentic to the show.