A friend (who is a Protestant Theology Student) and I discussed this issue.
The question is: Is God actually forgiving by Nature?
As with any question in Christendom there exists no unified beliefs and so I will need to answer it by Groups of churches/denominations/theologies.
Interestingly enough he isn't forgiving in the normal sense we imagine here, because Traditional Protestantism holds to the Penal Substitution Atonement model whereby God (The Father) pours out his wrath on God (The Son) on the Cross whereby the sins of the Elect are atoned for. Therefore, God's nature here is only wrathful or Just if you want to use a nicer word. He never actually forgives an offense that remains, he demands restitution for every wrong even if it is paid out by Jesus. In the 'balance' of sin and wrath he is equal, there is no need for a forgiving nature because he's already dealt with it.
This is actually a yes and no answer.
Yes: Jesus didn't completely atone for your sins past, present, and future, as he does in Protestantism, it's more like you start becoming just and then if you're good enough he lets you in anyway after purging you a bit.
But at the same time, No, he isn't forgiving, because he's still making people pay for their lack of sanctification/holiness with purgatory, etc. So God is a little more forgiving in Catholicism, but still not very forgiving, not as forgiving as most humans are. Ex. If my friend Dan lies to me, I don't think an adequate reparation would be a lifelong torment.
The Arians/Unitarians/Universalists/Liberals actually end up being the only ones with a God who is truly forgiving. Jesus death is usually viewed in these groups as an unfortunate mistake or as a supreme act of love in the Lombardian sense. God actually looks at people who are sinful and 'not good enough' and says 'I forgive you', he actually forgives them.
I'm not arguing for Liberal/Modernist Christianity, I'm a Roman Catholic, but I still think it's interesting that in my definition of the terms, they're the only ones who have a God that actually forgives, that actually 'gets over it'.