Plums (Prunus domestica); red currants (Ribes rubrum); black currants (Ribes nigrum); gooseberries (Ribes uva-crispa); sour cherries (Prunus cerasus); pumpkins (Cuccurbita spp.) are sources for valuable fruit- and berry-juice and cider production. This process leaves a large number of by-products (BP) in the form of pomace, which accounts for up to 80% of the raw material. This by-product represents a rich source of biologically active compounds, especially in the form of different pectic polysaccharides. The pectin extracted from commercial fruits such as citric fruits and apples has high medicinal properties, can be used as edible films and coatings, and is also useful in texture improvement and gel production in the food industry. However, many under-utilized fruits have received little attention regarding the extraction and characterization of their high/value pectin from their by-products. Moreover, the commercial extraction process involving strong acids and high temperature to obtain high-purity pectin leads to the loss of many bioactive components, and these lost components are often compensated for by the addition of synthetic antioxidants and colorants. The aim of the research is to extract pectin from juice production by-products with hot-water extraction using weak organic (0.1 N) citric acid, thus minimizing the impact on the environment. The yield of pectin (PY = 4.47–17.8% DM), galacturonic acid content (47.22–83.57 g 100−1), ash content (1.42–2.88 g 100 g−1), degree of esterification (DE = 45.16–64.06%), methoxyl content (ME = 4.27–8.13%), the total content of phenolic compounds (TPC = 2.076–4.668 µg mg−1, GAE) and the antiradical scavenging activity of the pectin samples (DPPH method (0.56–37.29%)) were determined. Free and total phenolic acids were quantified by saponification using high-pressure liquid chromatography (HPLC). The pectin contained phenolic acids—benzoic (0.25–0.92 µg mg−1), gallic (0.14–0.57 µg mg−1), coumaric (0.04 µg mg−1), and caffeic (0.03 µg mg−1). The pectin extracts from by-products showed glucose and galactose (3.89–21.72 g 100 g−1) as the main neutral sugar monosaccharides. Pectin analysis was performed using FT-IR, and the rheological properties of the pectin gels were determined. The quality of the obtained pectin from the fruit and berry by-products in terms of their high biological activity and high content of glucuronic acids indicated that the products have the potential to be used as natural ingredients in various food products and in pharmaceutical products.