Background: The COVID-19 pandemic exposed substantial shortcomings in emergency hospital systems, demonstrating that conventional linear models fail to accurately depict the intricate interdependencies among clinical, operational, and infrastructural components. This research employs nonlinear and fractal analytical frameworks to characterize and forecast the dynamic behavior of the principal emergency clinical hospital in Iași, Romania, for the years 2020–2022. Methods: The study employed a mixed-methods approach, combining retrospective analysis of hospital risk factors (infection rates, staff absenteeism, and ICU occupancy) with computational simulations. We analyzed time-series data using several nonlinear metrics, including the Higuchi fractal dimension, Lyapunov exponents, Shannon entropy, and multifractal detrended fluctuation analysis (MF-DFA). A modified Lotka–Volterra system was utilized to simulate feedback mechanisms between clinical and operational stressors. Results: The analyzed variables demonstrated nonlinear dynamics and fractal self-similarity across temporal scales (Hurst exponent = 0.73 ± 0.04; D_H = 1.35–1.49). Positive Lyapunov exponents (λ > 0) indicated deterministic chaotic behavior, whereas the multifractal spectrum width (Δα ≈ 0.62) indicated heterogeneous scaling patterns across hospital departments. Increasing fractal complexity was closely associated with diminished clinical efficiency and staff availability. Conclusions: The progression of pandemic-related risk followed a trajectory consistent with self-organized criticality, transitioning from stable to oscillatory and ultimately to chaotic phases as system load intensified. Adding nonlinear markers such as fractal dimension, Lyapunov exponents, entropy, and multifractal parameters to hospital monitoring systems could improve early warning systems and make it easier for healthcare organizations near critical thresholds to manage risk
