A GAGG:Ce crystal scintillator contaminated by thorium was used to search for double-$\alpha$ decay of ${}^{224}$Ra. Considering that the daughter nucleus is ${}^{216}$Po with the half-life of 0.144 s, the time-amplitude method was utilized to search for ${}^{224}$Ra $2\alpha$ events, which allows for a significant background reduction in the region of the expected effect. No double-$\alpha$ decay events were observed, allowing to set a lower limit on the decay half-life, $lim{T}_{1/2}=64$ yr at 90% C.L., which exceeds the theoretical predictions of the symmetric model of $2\alpha$ decay ($T_{1/2}=2.6$ yr).