After an interview with a company they offered me a salary matching the one I asked for. However when considering the other benefits I was getting at the current employer (which I wasn't getting at from the other company) I realized my initial request was too low and I asked for more. They agreed to my demand and I accepted this new offer via email and telephone, because I was sure my current employer wasn't going to match it.
When I informed my current employer of the offer I received they came back with a counter offer which is slightly better and is actually a 60% increase from what I am currently getting from them.
Now I would like to ask the other company (not my employer) that I need some time to reconsider their offer even though I already informed them that I accept it because I got a counter offer from my current employer that is very competitive.
Is this a bad idea? And how should I proceed in asking for this additional thinking time while being professional and polite.