Can I be fired and if so, can I take action against the employer?

My ex coworker recruited me to his company where he is a respected manager. A section of the online app asked about education and I tried to enter that my level is some college and I am going to enter a program at a university for a bachelors this spring. I had a difficult time with the website not working properly, it entered that I had a bachelors there. Thought I fixed it. I called the friend that referred me for 3 days telling him I was having a hard time getting my app right. My resume didn't state I had any college and wasn't mentioned in interview and they decided to give me a much higher position (that didn't require degree) instead of entry level. Background check company called them after I worked there two weeks and told them I falsified info and are giving me a hard time.