Rare earth (RE)-based MgCu$_2$-type intermetallic compounds are studied by first-principles calculations to search their spintronic applications and thermodynamic stability. We found in our observations that both EuPt$_2$ and GdPt$_2$ have half-metallic properties. The band gap arises in spin-down channel which is found to be enhanced when we move from EuPt$_2$ (0.91 eV) to GdPt$_2$ (1.04 eV). From magnetic properties we got to know that the magnetic moments of RE and Pt are opposite in direction and total magnetic moment is found to be 7.01 $\mu_B$ and 4.1 $\mu_B$ for EuPt$_2$ and GdPt$_2$ respectively. Presence of $5d^1$ electron in Gd valance shell is the reason for this difference in magnetic moments. No negative frequency has been observed in the phonon dispersion curve of both systems indicating the dynamic stability of MgCu$_2$-type structure. Several thermodynamical potentials are also explored in the present study using quasiharmonic approximation. The half-metallicity of dynamically stable REPt$_2$ systems is appropriate for spintronic applications.