Doubtful. Beeper has several legitimate causes of action to bring their own suit, if they really expect that outcome (and more importantly, if they have the financial resources to litigate)
However, the regular pattern we've seen is that companies use copyright and/or ToS as basis for C&D'ing (with threat of litigation) developers that produce adversarily-interoperable solutions.
If Apple did so (and Apple would've absolutely done it if Beeper wasn't a reasonably well-funded adversary), Beeper would suddenly have an argument, as well as the support of the media ("Apple sues small company for opening up iMessage to Android") and the potential to establish a legal precedent that would threaten not just Apple but the tech industry at large.
However, it's a common pattern that large companies can shut down adversarially-interoperable projects by threatening litigation against the developers. The lawsuit might be baseless but would still require upfront resources to defend; this is what these companies rely on, so they get their way without the argument ever getting into a courtroom.
If Apple brought forward such a lawsuit and Beeper actually litigated it to the end (and actually got it into a courtroom), it would risk creating a legal precedent that would enshrine adversarial interoperability as legal and make such future bullshit legal threats ineffective. That is a major risk not just for Apple but the tech industry at large.
However, it seems that Beeper effectively got what they wanted (bipartisan calls for regulatory action against Apple, and lots of media coverage over the issue) without any lawyers being involved.
Media attention, yes. Policy support, no.
Beeper could definitely be prosecuted by the Feds.
Aaron Swartz is probably the most famous example of someone being prosecuted using the Computer Fraud and Abuse Act. He was merely accessing a web server without permission and wasn't even trying to turn a profit.
There are many instances of "adversarial interoperability" (somebody else already mentioned screen scraping of online banking for budget management tools already in a different thread), and I haven't seen the CFAA being thrown at the responsible parties all that often.
I'd be quite curious to see precedent being set here, but I doubt it'll happen. Apple has much more to lose than to gain from that:
They can play cat and mouse on the tech side as long as they want, but with all the attention and scrutiny of a lawsuit, I could see a small chance of Apple ending up having to open up their service for interoperability.
If they're willing to prosecute some kid who wasn't even trying to make a profit off of his access to a web server, why wouldn't they prosecute a company for trying to sell hacked access to someone else's servers?
Also, there have been many prosecutions under this law. Aaron's case is just the most infamous example.
Beeper is trying to use an api to send message to users. They are not getting nor trying to get shell access to apple servers.
Beeper intent is to serve both Apple customers and non Apple customers to exchange messages securely. Its goal is interoperability, not stealing, or blindly using resources it doesn't own.
The intent is to sell hacked access to somebody else's servers.
If I sell hacked access to Microsoft's Office 365 servers, I can claim to have any motivations I like. It's still a crime.