The NFL can prohibit X from using its trademarks, or markings that would be likely to confuse a reasonable person, in any of X's business marketing.
The NFL's shield wordmark has USPTO registration number 4682232 (deep links aren't allowed, but their search engine is here.) The images you've added don't seem alike enough to be confusing to me (I don't think I'd be mistaking one for the other), but in a lot of settings the app name will have only text, not an image. Then you get at the trademark issue for the term "NFL" itself.
NFL can trademark that three-letter sequence, and there are good reasons for them being able to keep control of that trademark to authenticate sourcing. For example, if I said "this hamburger is from McDonalds" you have certain expectations about it based on the brand, and McDonalds the company has a legally protected interest in preventing me from saying that if it wasn't true. I have no right to put that 9-letter sequence on my burger joint without a license from the corporate giant.
Similarly, The NFL has a legally protected interest in preventing somebody from advertising or implying that an information source or the information contained therein is licensed and/or an official source of information, if that's not true. A reasonable person might expect that an information source about NFL player injuries, with "NFL" in its name, would be an official licensed source of that information; the real NFL can seek an injunction and/or penalties if that's not actually the case. "Pro Football Player Injury Tracker" might not carry the same expectation.
At the bottom of NFL.com, they claim, emphasis added, that "NFL and the NFL shield design are registered trademarks of the National Football League." So they have put folks on notice they intend to defend that three-letter sequence as theirs, at least for all subjects related to professional American football.
Sean Mullin says "The NFL is notorious for guarding its intellectual property rights to the full extent of the law." X should probably spend some time with an IP lawyer.
If you're asking if it's possible for X to legally use the NFL's trademark, the short answer is yes. It is possible for X to license use of the trademark from the NFL so long as the NFL is willing to grant/sell the license on terms that X also agrees to.
See this prequalification page linked to from this law firm's website, or click "Become a Licensee" here to see that in context.