Shape memory polymer composites (SMPCs) can be used to improve the mechanical properties of the shape memory polymers (SMPs). They also have the potential to enhance or enable new stimulus approaches and novel shape memory effects (SMEs). In this paper, a thermoviscoelastic finite deformation constitutive model is developed for thermally activated unidirectional continuous elastic fiber reinforced SMPCs. Since the structural relaxation and viscous flow mainly exist in the SMP matrix, an internal state variable modeling approach is used to describe the thermomechanical behavior of the SMPCs. Recent works mainly focus on the thermomechanical behavior of carbon fiber reinforced SMPCs in the small strain range, for the tensile tolerant strain of carbon fiber is small. The model that is developed here shows that the unidirectional continuous carbon fiber reinforced SMPCs with proper fiber inclination angle and volume fraction can also be used in finite deformation range for the first time. The finite deformation thermomechanical response of unidirectional continuous carbon fiber reinforced SMPCs with different inclination angles and volume fractions is addressed here. Therefore, this study provides useful guidance for reasonable design of unidirectional continuous elastic fiber reinforced SMPCs.