We present the first direct search for exotic Higgs boson decays $H\ensuremath{\rightarrow}\mathcal{AA}$, $\mathcal{A}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ in events with two photonlike objects. The hypothetical particle $\mathcal{A}$ is a low-mass spin-0 particle decaying promptly to a merged diphoton reconstructed as a single photonlike object. We analyze the data collected by the CMS experiment at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ corresponding to an integrated luminosity of $136\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. No excess above the estimated background is found. We set upper limits on the branching fraction $\mathcal{B}(H\ensuremath{\rightarrow}\mathcal{A}\mathcal{A}\ensuremath{\rightarrow}4\ensuremath{\gamma})$ of $(0.9--3.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ at 95% confidence level for masses of $\mathcal{A}$ in the range 0.1--1.2 GeV.