Can you and your fiance get married and go through with the proceedings for him getting his green card without you having to file for the fiance visa?

If one entered the US legally, i.e. was inspected and granted entry, one can apply to adjust status to that of a permanent resident through marriage to a US citizen while remaining in the US. However one should seek legal advice with a competent Immigration attorney who will be able to advise you based on your circumstances. The application can be denied if it is deemed there was immigrant intent on entry to the US.

You can also find a great guide to adjusting status with concurrent filing of an I-130 at http://www.familybasedimmigration.com/forum/showthread.php?t=271 It also contains lots of links to official sources at uscis.gov regarding relevant information.