The American Basketball Association was formed as a competitor to the National Basketball Association with the aim of forcing the NBA to merge with them and absorb the ABA franchises into the league. In 1976, nine years after the leagues formation, four franchises (Spurs, Pacers, Nets, and Nuggets) joined the NBA.

In rugby, there are two major "codes", rugby league and rugby union, that have separate leagues and slightly different rules. The split between the two codes was caused by certain teams wanting to move toward professionalism. The breakaway league that those teams began was the start of rugby league, while rugby union retained its amateur standing until 1995.

I don't think either of these perfectly fit your question, as neither involved the creation of a league to compete with an already professional that went on to be successful in the long term. However, looking into these examples should be useful for you.

You could argue that NASCAR is such a thing. The premise would be to say that USAC is the official, premiere auto racing sanction body, because they sanctioned the US GP. Or was it the AAA? Or SCCA? What about CART? Lot's of stuff going on there in racing history!

Now, if we maintain that whoever sanctiones the US GP, the official event of the worldwide, official sanction body FISA which is the motor sports arm of the official, international, global, traditional automobile federation FIA, if that is the premier thing to which we are in search of an sucessful alternative, then at least NASCAR if not others as well has to count as such an alternative.

We can argue the details, to come to an understanding whether it is a true example or not. It's quite an intersting question since outside the US, pro sports is usually organized by institutions and not corporations.

I would say NASCAR is a positive answer to your question.

Would MLS count as an alternative to The European football leagues?

On the same sport, the Premier league in the UK was set up by the largest clubs on top of the existing divisions.

Professional baseball has a history of leagues competing and eventually merging with each other. The National League was formed in 1876 and faced competition from the American Association starting in 1882 and the American League in 1901.

The American and National Leagues were legally distinct organizations up until 2000. All teams are now under the umbrella of "Major League Baseball".

Does anyone have knowledge about boxing? Seems there are multiple boxing organizations, the ones I hear most about are the WBA, WBC, and WBO, but I know there are tons of others. I'm not even sure if they are organized like leagues.