Large language models (LLMs) can provide rich physical descriptions of most worldly objects, allowing robots to achieve more informed and capable grasping. We leverage LLMs' common sense physical reasoning and code-writing abilities to infer an object's physical characteristics$\unicode{x2013}$mass $m$, friction coefficient $μ$, and spring constant $k$$\unicode{x2013}$from a semantic description, and then translate those characteristics into an executable adaptive grasp policy. Using a two-finger gripper with a built-in depth camera that can control its torque by limiting motor current, we demonstrate that LLM-parameterized but first-principles grasp policies outperform both traditional adaptive grasp policies and direct LLM-as-code policies on a custom benchmark of 12 delicate and deformable items including food, produce, toys, and other everyday items, spanning two orders of magnitude in mass and required pick-up force. We then improve property estimation and grasp performance on variable size objects with model finetuning on property-based comparisons and eliciting such comparisons via chain-of-thought prompting. We also demonstrate how compliance feedback from DeliGrasp policies can aid in downstream tasks such as measuring produce ripeness. Our code and videos are available at: https://deligrasp.github.io