Impedance cardiography (ICG) is a rapidly growing non-invasive cardiac health monitoring approach. Synchronous detection of ICG requires an FIR filter to remove the high-frequency carrier signal. Low power consumption and compact area are critical considerations in the design of portable biomedical systems. This paper proposes a novel product quantization-based optimization strategy for the Urdhva Tiryagbhyam Sutra-based multiplier architecture. This paper presents an ASIC design of a low-power and low-area 64th-order programmable FIR filter architecture using the optimized Urdhva Tiryagbhyam Multiplier. The programmable architecture empowers medical practitioners to select the carrier frequency at which the ICG analysis will be performed. The elimination of redundant multipliers from the design based on the filter coefficients is demonstrated. The programmable Vedic FIR filter architecture (described in VHDL) is implemented on the Basys-3 FPGA board for rapid prototyping. The RTL-to-GDSII flow has been completed using Cadence digital design and sign-off tools for the SCL-180 nm technology. The results indicate that the proposed filter architecture occupies 41.33% less area and 42.16% lower power consumption than the contemporary designs described in the literature.