I'm so, so sorry for your loss and for the challenges this new season is bringing your family. It has always baffled me that even after having been married a long time, they could re-marry (or date) right away. It just seems to fly in the face of our womanly logic: how could you just move right on if your wife meant so much to you??
My dad, who is a counselor, explained that for men it simply isn't that....well...simple. As men are MUCH more able to compartmentalize their feelings, they can separate the love they had for their wife with the appreciation for a new companion. He also said that, typically, the longer men were married before, the faster they marry after their partners pass because they are so accustomed to having someone to care for them - they don't do well alone.
I know that wasn't your question, but I wanted you to see that your father isn't particularly insensitive - this is typical.
As for her, I would think that someone who truly cared for him would never attempt to put his family members at odds with each other. She should understand that this is difficult and be willing to take things slowly and carefully - out of respect for his family. I do think she was insensitive to not defer to your plans for your father and give you all some space on Father's Day. You can't change her, though and what's done is done.
Moving forward, I would suggest to just be as open and honest with your dad as possible, but in a non-accusing, non-confrontational way. If he is like most men, you won't get far that way. Acknowledge HIS feelings in this as well as your own: that you understand how hard it must be for him and you are happy if he is happy, but you're still struggling and - honestly - it wouldn't matter who the new person was. Your feelings are not necessarily about her personally, just the fact that she is a replacement and that's hard. It takes time, so just ask him to be patient with you while you try to work through this - and for his sake, you MUST follow through and try if you want to preserve your relationship.
good luck to you...