GRBs have long been considered as potential sources of hadronic acceleration of ultra-high energy cosmic rays and, more recently, as potential sources of the diffuse neutrino flux measured by IceCube. This thesis presents a search for neutrinos correlated with 2,091 gamma-ray bursts (GRBs) in prompt and extended time windows using data from the IceCube Neutrino Observatory ranging from May 2011 to October 2018. Ten time windows, ranging from 10 seconds to -1/+14 days around the time of the GRB's gamma-ray emission, were searched for coincident neutrino emission and a p-value was assigned based on the most significant time window. The results for all 2,091 GRBs were divided by region of the sky and observed gamma-ray emission time, and were evaluated with a Binomial test, which was found to be consistent with background. The 23 most significant GRBs were examined in more detail and they were also found to be consistent with background. Limits were set assuming two flux models: equal neutrino flux at Earth for every GRB and standard candle neutrino emission. The equal flux at Earth model led to similar constraints as previous IceCube prompt GRB studies, namely that GRBs are responsible for, at most, a few percent of the diffuse flux. The results of the standard candle analysis, however, indicate GRBs may be contributing up to 11% of the diffuse neutrino flux up to 1,000 second timescales, which leaves the door open to GRBs as neutrino sources and hadronic accelerators of at least some of the ultra-high energy cosmic rays.