A search for a new scalar particle decaying to a pair of photon-jets---collimated groupings of photons---is performed with the ATLAS detector at the Large Hadron Collider. The dataset of $pp$ collisions at a centre-of-mass energy of $13~\mathrm{TeV}$ was collected in 2015 and 2016, corresponding to an intergrated luminosity of $36.7~\mathrm{fb}^{-1}$. Photon-jets arise when a high-mass scalar particle decays into a pair of light particles, which consecutively decay to photons. When these light particles are highly boosted, their decay to photons lead to photon-jets. Candidate events with a pair of photon-jets are searched for in events containing two reconstructed photons with high transverse energy. No statistically significant excess of events from the background expectation is observed. The results are interpreted in the scenario of a scalar particle with a narrow width ($X$) decaying into a pair of spin-0 particles ($a$). Upper limits are placed on the product of the cross-section and branching ratios $\sigma \times {\mathcal {B}}(X \rightarrow aa) \times {\mathcal {B}}(a \rightarrow \gamma \gamma)^{2}$ and $\sigma \times {\mathcal {B}}(X \rightarrow aa) \times {\mathcal {B}}(a \rightarrow 3\pi^{0})^{2}$ for the ranges $200~\mathrm{GeV}<m_X<2~\mathrm{TeV}$ and $m_a<0.01\times m_X$.