When someone infriges on your branding, copyright, patent, or other intellectual property, your only legitimate recourse is taking action in the court system.   
So yes, a judge will decide.     
I don't see why this is a problem.     Should J&J roll over after spending $$$$ to create, promote and build their branding?     
The American Red Cross is now competing with J&J in a commercial venture.     The ARC has to play by the rules like everyone else.   If a judge decides the American Red Cross crossed the line, so be it.   I doubt J&J would have taken on this PR nightmare if they weren't convinced that they have a good chance to prevail.   
IMO, the emotional argument presented by the American Red Cross's CEO was telling.  
If the law was on their side, they could just argue that position.    I haven't seen anything presented by the ARC that would indicate that they don't see a problem with their use of the branding.   The only thing I could find is how the profit hungry corporation is trying to damage a benevolent Not-for-Profit.
It will be interesting to see how this plays out.   Thanks for the post.
  
   
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Gary