The last time I saw him was when I was 7 years old. I'm 15 years old now. He stood me up on father's day and I've never spoken to him again My mother has made numerous attempts to get him to be a part of my life. She left her Facebook account up a few years ago and I saw a message she had sent to him asking of he wanted to talk to me. His reply was that he wanted a DNA test because he didn't think I was his. I have half-siblings on his side and I look just like my older sister and youngest brother. And I look a lot like him. My mother told him to arrange it, but I never asked her about it and it was never done. I found him myself on Facebook two weeks ago and sent him a message and a friend request. He blocked me. I asked my mom to search for his name under her account and he showed up, but he doesn't show up for me. I just hate him now! I want to know my siblings and it seems they are all a part of his life. Why doesn't he want me?

Heart jerking story but...
Can I sue my biological father for emotional distress? NO.

Before you could even contemplate this cause of action you must get a DNA test done.
Just because you share some features with siblings is meaningless.

No, you can't sue for emotional distress.
I know this hurts. Hate is just a way of dealing with the hurt of being unloved. You are old enough now to realize this, and to realize that this has nothing to do with you and everything to do with your mother. You probably don't know why he is so angry at her as to deny you. Maybe you can sit down with her and gently ask her.
What I'm wondering about is child support - she never got any? Never tried? The court would have ordered DNA testing most likely.
Be very, very careful about how you pursue this, because you could be wrong, and that will hurt too. 15 is old enough for some things and not others.

Is your mom collecting child support from him at this time or is there an order from the court for it?

Yes, my mom collects child support from him occasionally (usually at tax time). She tells me when she gets it and I have all the info to check it myself if I want, she puts half in my college savings and either uses the other half for bills if we need it at the time or takes me shopping. When I say I have similarities to my siblings, it's not just a little resemblance. I look just like my older sister and so much like my sperm donor, from facial similarities to body type. My older sister and I could practically be twins and I actually have an online relationship with her, we also share many interests and talents. If there was any doubt about one of my parents being my biological parent, it would be my mom (lol). We look nothing alike. As far as DNA testing, it's never been done even though he said he wanted it. My mom told him to arrnge the testing and we would be there but he never stepped up. My mom has always been honest with me about her own faults and the circumstances with my dad. While she wasn't perfect in the relationship, he was emotionally abusive and became an addict before my mom packed up and left. He looked down on her because of where she was from and because she wanted to move back to a small town. She has a very southern accent, which many of people mistake for ignorance, when in fact she is well educated and very articulate and I know this sounds crazy coming from a teenager, but my mom is one of the most intelligent people I know, but not one of those condescending types. She and I volunteer in our free time helping adults learn to read and she teaches a class helping people get their GED. She's just amazing.
Anyway, the more I've thought about it and after talking to mom, the less I care. He's the one losing out on my awesomeness and as my younger siblings grow older I can seek relationships with them if they're interested. I know the full names of the three other mothers and kids I'm related to so I'll just bide my time and will be there if any of them want to know me.

Honestly that does not mean you are his, I have an adopted son, he looks more like me than any of my other boys ( I have 5 boys) Everyone says he looks like me, acts like me and more. So without a DNA test you don't know for sure.

And your dad ( or this man) may have real reasons to not be sure, I am sorry but those things happen.
Now the issue I see, he has to be declared legally your father either by DNA test or birth certificate or something for him to be ordered to pay child support.
Why does not your mom not go after him harder for the money?

You can't sue him for emotional damage.

It probably hurts that he doesn't want you in his life, and I don't know what you're dealing with, but I know that I was like him, except I was the child. My biological dad contacted me and he wanted a relationship and was totally ready to be part of my family, but DNA does no build a relationship, DNA doesn't make things change and when he hurt me over and over I ended it... but I don't know, I just guess I kind of understand the whole not wanting someone to be in your life, and that I don't consider him family even if we have the same DNA...