In this study, giant kelp was explored under various conventional and ultrasound-assisted extraction (UAE) conditions for the extraction of protein, its hydrolysis, and ultrafiltration to generate multiple fractions. The amino acid composition of all the fractions and their biological activities in vitro, including angiotensin-converting enzyme I (ACE) inhibitory activity and antioxidant activities (2,2-diphenyl-1-picrylhydrazyl (DPPH) radical scavenging, reducing power (RP), and ferrous chelating (FC) activities) were tested by storing the compounds for 2 weeks at various temperatures (−20–60 °C) and pHs (2–11) to elucidate their thermal and ionic stability, respectively. The yield of protein extraction using the conventional method was lower (≈39%) compared to the use of UAE (150 W, 15 min), which achieved protein recoveries of approximately 60%. After enzymatic hydrolysis and ultrafiltration, low-molecular-weight (MW) hydrolysates had the highest levels of ACE inhibitory (80%), DPPH (84%), RP (0.71 mM trolox equivalents), and FC (81%) activities. Amino acids associated with peptides of high biological activities, such as Val, Ala, Asx, Gly, Lys, Met, Leu, and His, were at higher levels in the low MW fraction compared to any other sample. The biological activities in vitro of all the samples fluctuated under the multiple storage conditions studied, with the highest stability of all the samples appreciated at −20 °C and pH 7. This study shows for the first time the use of giant kelp as a promising source of bioactive peptides and indicates the optimum processing and storing conditions for the use of these compounds as nutraceuticals or functional foods that could help in the prevention of cardiovascular disorders and multiple chronic diseases associated with oxidative damage.