The vast majority of real-world patient information resides in unstructured clinical text, and the process of medical abstraction seeks to extract and normalize structured information from this unstructured input. However, traditional medical abstraction methods can require significant manual efforts that can include crafting rules or annotating training labels, limiting scalability. In this paper, we propose UniMedAbstractor (UMA), a zero-shot medical abstraction framework leveraging Large Language Models (LLMs) through a modular and customizable prompt template. We refer to our approach as universal abstraction as it can quickly scale to new attributes through its universal prompt template without curating attribute-specific training labels or rules. We evaluate UMA for oncology applications, focusing on fifteen key attributes representing the cancer patient journey, from short-context attributes (e.g., performance status, treatment) to complex long-context attributes requiring longitudinal reasoning (e.g., tumor site, histology, TNM staging). Experiments on real-world data show UMA's strong performance and generalizability. Compared to supervised and heuristic baselines, UMA with GPT-4o achieves on average an absolute 2-point F1/accuracy improvement for both short-context and long-context attribute abstraction. For pathologic T staging, UMA even outperforms the supervised model by 20 points in accuracy.