Generative AI has demonstrated strong potential in healthcare, from clinical decision support to patient-facing chatbots that improve outcomes. A critical challenge for deployment is effective human-AI communication, where content must be both personalized and understandable. We introduce MedReadCtrl, a readability-controlled instruction tuning framework that enables LLMs to adjust output complexity without compromising meaning. Evaluations of nine datasets and three tasks across medical and general domains show that MedReadCtrl achieves significantly lower readability instruction-following errors than GPT-4 (e.g., 1.39 vs. 1.59 on ReadMe, p<0.001) and delivers substantial gains on unseen clinical tasks (e.g., +14.7 ROUGE-L, +6.18 SARI on MTSamples). Experts consistently preferred MedReadCtrl (71.7% vs. 23.3%), especially at low literacy levels. These gains reflect MedReadCtrl's ability to restructure clinical content into accessible, readability-aligned language while preserving medical intent, offering a scalable solution to support patient education and expand equitable access to AI-enabled care.