So If Merkel Retires As Chancellor Because of Bad Health

This is a short essay and simply lays out the way that the German system would work in this case....because of health issues, Merkel resigns.

The CDU Party will meet and do a in-house committee to put forward a couple of names of 'acceptable' candidates to fill the role. 

They will ONLY be forced to have a out-of-cycle election (next would be fall of 2021 normally) IF the SPD Party (their coalition partner) says 'NO' to the candidate put up.  It's possible that the CDU committee might have a short discussion with the SPD, to see if there is any 'no-go' candidate on the list....but I seriously doubt that the CDU will care about that approval. 

However, this is not the end of the world....the CDU Party could accept the 'NO' answer to their selection, and seek a coalition with the FDP and Green Party.  If they accepted this new Chancellor and the partnership deal, then things would continue to work until the fall of 2021, when the next election occurs.

If neither coalition works, then an election would be called for....within about 90 days.  In this scenario, the SPD Party is hurt more than expected because they've lost public support since the last election (2017).  The odds are heavily in favor of the SPD agreeing to the candidate announced.

Who would vote within the CDU process for the Chancellor candidate?  Anyone who is a member (paying-member) of the party.

Most people will say it's a short list of four to six people who would 'fit'.

The positives and negatives of this period?

First, most Germans are in agreement that a new Chancellor might be very positive (they've said this for the past three years). 

Second, the Green Party has gained at least twelve points since the last national election.  They probably would prefer a weaker candidate for the CDU replacement....to continue their trend.

Third, yes, it's possible that this gets into a serious debate within the CDU Party, and they pick a place-holder to temporarily hold the job till 2021.

Fourth and final, if you expect a chaotic period....you might be surprised to see all of this replacement business done in thirty days or less.  It might actually be a one-star 'drama' at best. 

The public having any say in the replacement?  No.  It's a party matter.  If you paid to be a member of the party....you get a say....otherwise, its not of your business.