We investigate the thickness-dependent electronic properties of ultrathin SrIrO_{3} and discover a transition from a semimetallic to a correlated insulating state below 4 unit cells. Low-temperature magnetoconductance measurements show that spin fluctuations in the semimetallic state are significantly enhanced while approaching the transition point. The electronic properties are further studied by scanning tunneling spectroscopy, showing that 4 unit cell SrIrO_{3} is on the verge of a gap opening. Our density functional theory calculations reproduce the critical thickness of the transition and show that the opening of a gap in ultrathin SrIrO_{3} requires antiferromagnetic order.