Polymeric nanomaterials that degrade in acidic environments have gained considerable attention in nanomedicine for intracellular drug delivery and cancer therapy. Among various acid-degradable linkages, spirocyclic acetals have rarely been used to fabricate such vehicles. In addition to acid sensitivity, they benefit from conformational rigidity that is otherwise not attainable by their non-spirocyclic analogs. Herein, amphiphilic spirocyclic polyacetals are synthesized by Cu-catalyzed alkyne–azide “click” polymerization. Unlike conventional block copolymers, which often form core–shell structures, these polymers self-assemble to form core amphiphilic assemblies capable of encapsulating Nile red as a hydrophobic model drug. In vitro experiments show that while release from these materials can occur at neutral pH with preservation of their integrity, acidic pH accelerates efficient cargo release and leads to the complete degradation of assemblies. Moreover, cellular assays reveal that these materials are fully cytocompatible, interact with the plasma membrane, and can be internalized by cells, rendering them as potential candidates for cancer therapy and/or drug delivery.