Apply for green card while on ESTA

Yes, you could adjust status provided that you had no intention of staying permanently in the United States when you entered the country. The fact that you purchased a return ticket prior to your entry and you had a job lined up back home would help establish that you only intended to visit. Some of us handle these types of cases for a very reasonable flat rate fee for start to finish representation. Consider working with counsel so your case will process smoothly. 

If your girlfriend is a US citizen and if it was not your intention to immigrate at the time that you came into the US with the ESTA, it is possible to adjust status to permanent residence assuming that you have no grounds of inadmissibility. Due to the limitations of the Lawyers.com Forums, Alan Lee & Arthur Lee, Esqs.’ (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.