So I have accepted an offer. I was not expecting my current employer to counter with such a big offer. Can I bring that up to the company I accepted the offer from already to negotiate a better salary?
I believe that using this as a leverage would damage your reputation from the perspective of the new company. It would show that you can easily break your agreements (you have already made a mutual decision in terms of salary). Also, hiring is expensive (training, adapting, etc) and the employer hopes that you'd stay for as long as possible. If you would come back with new demands then chances are that you would be leaving them soon and cause them problems with replacing you, time spent on your training wouldn't pay off, etc.
I've been in this situation and I know how tempting that is but you are better off not doing that :)
Regarding accepting a counter-offer:
Technically you can. Is it a good idea? Generally not.
See, your current employer already knows that you want to leave. They are offering a raise not because they suddenly realized how great you are but because they don't have a replacement right now. Giving you a raise to stay buys them some time to find a replacement. Once they have found it, they have very little reason to keep you there because chances are, you will try to leave again soon.
The only exception might be if you have expertise in some niche that very few people do. At the end of the day, everyone is replaceable.
Also, you made the decision to leave them so probably there is something else other than money that was bothering you. If it was only money and they rejected a raise, that just supports my previous point.
Generally it's not a good idea to bring it up after you have already agreed on something. As an employer, people that go back on their word is a warning sign. It might just lose you the offer from your potential new employer.
What you could do, is mention that you did get a much better offer from your current employer but that you like the new company more and need a few days to think and talk things through with your significant other / a friend before you make a decision. That just might get your new employer to raise their offer without you having to go back on your word.
If you have accepted an offer, you have accepted an offer. The situation is that you will work for your old employer until the end of your notice, but you have mentally switched sides. You are looking forward to your new job. If your old company now makes a higher offer, that is just annoying, they should have offered it before you looked for a new position if they think you are worth it. (And indeed it is a good rule never to go back and take the higher offer, because you will be a marked man. Except if you are a marked woman. Or just marked).
What you can do is carefully tell the new company exactly that, that you received a higher offer, and how annoying it is that they waited until you decided to leave. No mention that you want more money, or that you are excepting more. Be genuinely annoyed with your old company. The result should be that your reputation with the new company will improve, and there is a chance of an improved offer.
Andreas' answer is very similar but a bit more aggressive, which increases both the chances of getting a better offer, and the risk of annoying your new company and losing out completely. Your choice.