Kevin:
As the executor of Jon's will, you have legal standing as to accounting for his possessions, assets and liabilities and legal claims for or against him, until the will is probated by the courts.
You couldn't distribute any of his things even if you wanted to, until the court determines the validity of his will and then distributes the assets accordingly.
If his father chooses to contest the will, he has that right. But if there is no real money involved, he isn't likely to do so.
I know how upset you must be so soon after his death and your reaction to his father is understandable. But I would also encourage you, after you have had time to deal with your grief, to try and understand that the man is his father, albeit a bad father, and you may want to show him some kindness and understanding in the future.
Of course you are under no obligation to do so, but as a father, no matter how displeased and disappointed I might be at my children or they in me, there is still a deep feeling for them that I will carry to my grave.
May you have peace!