We present SNIascore, a deep-learning based method for spectroscopic classification of thermonuclear supernovae (SNe Ia) based on very low-resolution (R $sim100$) 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 public 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 up to $90%$ of the low-resolution SN Ia spectra obtained by the BTS. SNIascore simultaneously performs binary classification and predicts the redshifts of secure SNe Ia via regression (with a typical uncertainty of $<0.005$ in the range from $z = 0.01$ to $z = 0.12$). For the magnitude-limited ZTF BTS survey ($approx70%$ SNe Ia), deploying SNIascore reduces the amount of spectra in need of human classification or confirmation by $approx60%$. Furthermore, SNIascore allows SN Ia classifications to be automatically announced in real-time to the public immediately following a finished observation during the night.