This is a big distinction and in reality the term accepting bitcoin will not go mainstream. Instead merchants will become "digital payment enabled". Which is basically the same thing however instead of accepting bitcoin directly they will be enabled to accept payment from ANY app that runs on bitcoin.
In the near future there will be two types of merchants, the ones who only accept cash and debit/credit cards, and the ones who accept those methods along with accepting "digital payments".
Accepting digital payments will mean accepting payment from ANY app that runs on bitcoin. This would mean any client side bitcoin wallet, any centralised wallet like coinbase, but more importantly they will also by default accept payment from apps like circle and abra and whatever new systems spring up in the future.
A good analogy would be to consider an office that only accepts letters in the mail (paper cash) and fax's (credit/debit cards), vs an office that accepts those mediums but also accepts email from any email service provider operating out of any country (the equivelant of digital payments). Like is taken for granted today.
All of this will take place ON bitcoin and using bitcoin behind the scenes, bitcoin will be the underlying network that runs it, but merchants would have the ability to accept payment from any digital wallet in any currency.
What would be required on the merchant side is nothing more than a "point of sale app" that allows them to recieve funds from any wallet app using any currency, from bitcoin to the bitcoin pegged digital dollar or peso or whatever. This is possible as all digital payment apps will be using bitcoin as the underlying rails of their system.
This open interoperable world of digital money will mean that people will move from "accepting bitcoin" to being "digital payments enabled" meaning they can accept digital payments in any form from any wallet app that runs on the internet of money (bitcoin).
It wont be a matter of saying I accept circle or I accept abra or I accept bitcoin.. it will be a matter of saying I am "digital payments enabled", so you can pay me with any digital currency app in any currency, and the merchant will always recieve their nominated currency as payment whether thats bitcoin or a bitcoin pegged digital fiat currency.
With the amount of bitcoin related apps coming to market now like abra and circle and all the other bitcoin wallets, it would be an easy choice for merchants. They will move away from seeing bitcoin as a random currency they can accept, and instead they will see many hundreds of wallet apps with millions and millions of users, and all they need to do to accept payment from them all is download one point of sale app! This will be a no-brainer for merchants both online and in-store.
The days of "accepting bitcoin" will fade but the idea of becoming "digital payment enabled" will explode!
-"Do you accept bitcoin?"
-"No why would I, its illegal volatile and a scam, plus its not even real money"
-"Are you digital payment enabled?"
-"No, why? Is it used alot? How can I set it up? Is it easy to use? You mean I can accept payment from all those apps and all these millions of users just by downloading one single app??
You get the picture…