No, accelerometers detect acceleration in linear direction (up/down, left/right, forward/backward), while gyroscopes detect rotation (pitch, yaw, roll). Smartphones have both, to detect all six degrees of freedom.

Edit: Apparently, for smartphones, all these sensors are integrated into a single MEMS chip, one sensor for each degree of freedom, three accelerometers and three gyroscopes: https://youtube.com/watch?v=9X4frIQo7x0