The growing interests in Internet of Underwater Things (IoUT) put forward more demands for underwater acoustic (UWA) communication technologies. However, the conventional orthogonal frequency division multiplexing (OFDM) technology underperforms in terms of the resistance to carrier frequency offset (CFO), symbol time offset (STO), and applicability to multiple application scenarios. To remedy the deficiencies, this paper introduces the emerging universal filtered multi-carrier (UFMC) technology into UWA communications and devises a two-stage index modulated UFMC (TSIM-UFMC) system for performance improvement. In addition, a deep learning-aided signal detector termed as TSIMNet is proposed for the TSIM-UFMC system over multipath UWA channels. Specifically, at the transmitter, the TSIM-UFMC system adopts two-stage index modulation to characterize the different position information of data, pilot, and inactive subcarriers in each subblock for ameliorating the bit error rate (BER) performance under channel interferences. At the receiver, the TSIMNet detector exploits configurable deep neural networks to encapsulate the index demodulation, channel estimation, and constellation demodulation for optimizing the system structure. Furthermore, TSIMNet can be deployed to detect the received signal online for the TSIM-UFMC system after trained by offline data. Numerical results reveal that the proposed TSIM-UFMC system exhibits achievable advantages in combating CFO and STO, and the TSIMNet detector actualizes a near-optimal performance on BER with preferable computational complexity.