The mechanical properties of liposomes are critical for their efficacy as drug delivery vehicles. A liposome's stiffness is a composite property of its membrane and encapsulated core, but deconvolving these contributions is a significant challenge, as current methods rely on model-dependent assumptions. This study introduces a novel, experimental method to isolate the core's mechanical contribution. We fabricated giant unilamellar vesicles (GUVs) with either a simple buffer or a viscoelastic hyaluronic acid (HA) core and characterized them using atomic force microscopy (AFM). The core's contribution was isolated by subtracting the force response of buffer-filled liposomes from that of HA-filled ones. The contact point was identified from the change in data fluctuations. Our results demonstrate that the HA core dramatically increases the liposome's resistance to deformation. While the membrane stiffness is dominant in small deformations (up to 25 nm), the viscous core becomes dominant at larger deformations, bearing over 80% of the total load for deformations larger than 150 nm. This direct, model-independent approach provides a tool for understanding the biomechanics of filled liposomes and enables the rational engineering of liposomal drug carriers with mechanical properties precisely tuned for enhanced therapeutic performance.