Fluid-structure interaction (FSI) is an omnipresent phenomenon and presents itself in a range of natural, biological and engineering systems. Two-way interactions between a flowing fluid and a deforming solid occur in a variety of physical scenarios. Our primary motivation comes from ship-ice interaction in Arctic regions during collision of ships with large ice sheets, drifting of ice floes and crack propagation in ice. A fully coupled continuum mechanics analysis is required to capture the dynamic response of the marine vessel navigating ice-covered Arctic waters. Given the physical demands of the problem, we adopt a phase-field based fully Eulerian approach to capture the multiphase interactions in the system. We employ a stabilized finite element formulation and a partitioned iterative procedure to solve the unified momentum equation comprising solid and fluid dynamics coupled with the Allen-Cahn phase-field equation. The evloution of solid strain in the Eulerian reference frame is governed by the transport of the left Cauchy Green deformation tensor. This allows us to compute the stresses in the solid phase accurately. We introduce a contact force approach to handle smooth elastic-elastic contact based on the overlap of the diffused interfaces of two colliding bodies. The implemented model is validated against the classical Hertz model for smooth dry contact and sensitivity studies are carried out for the model parameters involved. We extend the model to resolve contact between multiple bodies with identical physical parameters immersed in a surrounding fluid. The developed approach obviates the need for solving multiple phase-field equations or reconstructing the solid boundaries at every time step. Using the developed framework, we demonstrate the collision dynamics between multiple immersed bodies of varying sizes. We also showcase scenarios where active motion control of a specific body allows us to replicate ship-ice interactions.