I currently own and run all of the online (and offline) marketing for a 15 year old PT practice. I've also owned and operated several other health and fitness related practices. So I'm very familiar with your situation.

Business expansion and exit strategy planning can be excellent reasons for any business to consider branding and other components of marketing. Re-naming is not something to be taken lightly so it's wise of you to consider the implications.

The short answer to your questions..
1. Will re-naming the business affect its branding?
It can if the name is congruent with your marketing strategy.

2. Will re-naming the business affect organic online ranking and search results?
Again - it can. The bigger concern might be your decisions regarding your URL and factors such as website optimization, keyword selection, and proper use of local search strategy and execution.

The bottom-line: it's worth the effort if the investment and effort have a high probability of helping you accomplish your specific goals (which you did not explicitly state) AND if it's a part of a solid business strategy. By itself re-naming is not likely to have a significant positive impact on your bottom line and if poorly executed could result in confusion to your patients (past, current and future) and a negative impact on ranking and search results.

Let me know if I can be of any assistance. I wish you the best of luck!