Morse code recognition plays a very important role in the application of human–machine interaction. In this paper, based on the carbon nanotube (CNT) and polyurethane sponge (PUS) composite material, a flexible tactile CNT/PUS sensor with great piezoresistive characteristic is developed for detecting Morse code precisely. Thirty-six types of Morse code, including 26 letters (A–Z) and 10 numbers (0–9), are applied to the sensor. Each Morse code was repeated 60 times, and 2160 (36 × 60) groups of voltage time-sequential signals were collected to construct the dataset. Then, smoothing and normalization methods are used to preprocess and optimize the raw data. Based on that, the long short-term memory (LSTM) model with excellent feature extraction and self-adaptive ability is constructed to precisely recognize different types of Morse code detected by the sensor. The recognition accuracies of the 10-number Morse code, the 26-letter Morse code, and the whole 36-type Morse code are 99.17%, 95.37%, and 93.98%, respectively. Meanwhile, the Gated Recurrent Unit (GRU), Support Vector Machine (SVM), Multi-Layer Perceptron (MLP), and Random Forest (RF) models are built to distinguish the 36-type Morse code (letters of A–Z and numbers of 0–9) based on the same dataset and achieve the accuracies of 91.37%, 88.88%, 87.04%, and 90.97%, respectively, which are all lower than the accuracy of 93.98% based on the LSTM model. All the experimental results show that the CNT/PUS sensor can detect the Morse code’s tactile feature precisely, and the LSTM model has a very efficient property in recognizing Morse code detected by the CNT/PUS sensor.