We present SNIascore, a deep-learning method for spectroscopic classification of thermonuclear supernovae (SNe Ia) based on very low-resolution (R~100) data. The goal of SNIascore is fully automated classification of SNe Ia with a very low false-positive rate (FPR) so that human intervention can be greatly reduced in large-scale SN classification efforts, such as that undertaken by the Zwicky Transient Facility (ZTF) Bright Transient Survey (BTS). We utilize a recurrent neural network (RNN) architecture with a combination of bidirectional long short-term memory and gated recurrent unit layers. SNIascore achieves a < 0.6 % FPR while classifying ~90 % of the low-resolution SN Ia spectra obtained by the BTS and needs no human supervision. SNIascore simultaneously performs binary classification and predicts the redshift of secure SNe Ia via regression. For a magnitude-limited SN survey (~70 % SNe Ia), deploying SNIascore reduces the amount of spectra in need of human classification or verification by 60 %, which results in a highly significant reduction in the man hours needed to run a spectroscopic survey.