Cosmic-ray interactions with the solar atmosphere are expected to produce particle showers which in turn produce neutrinos from weak decays of mesons. These solar atmospheric neutrinos (SAνs) have never been observed experimentally. A detection would be an important step in understanding cosmic-ray propagation in the inner solar system and the dynamics of solar magnetic fields. SAνs also represent an irreducible background to solar dark matter searches and a detection would allow precise characterization of this background. Here, we present the first experimental search based on seven years of data collected from May 2010 to May 2017 in the austral winter with the IceCube Neutrino Observatory. An unbinned likelihood analysis is performed for events reconstructed within 5 degrees of the center of the Sun. No evidence for a SAν flux is observed. After inclusion of systematic uncertainties, we set a 90% upper limit of 1.02^+0.20_−0.1810⁻¹³ GeV⁻¹cm⁻²s⁻¹ at 1 TeV.