Ultraviolet communication (UVC) serves as a promising supplement to share the responsibility for the overloads in conventional wireless communication systems. One difficulty lies in how to address the inter-symbol-interference (ISI) from the strong scattering nature and the time-varying channel response. This is more challenging for the energy-constrained scenarios (e.g., underwater UV), as existing coherent schemes (i.e., requiring exact channel information for signal detection) become less attractive given the computational and storaging complexity for repeated time-varying channel estimation and statistical signal detection. In this work, a novel non-coherent paradigm is proposed, via the exploration of the UV signal features that are insensitive to the ISI. By optimally weighting and combining the extracted features to minimize the bit error rate (BER), the optimally-weighted non-coherent detection (OWNCD) is proposed, which converts the signal detection with ISI into a binary detection framework, with a heuristic weight updating approach for time-varying channel. As such, the proposed OWNCD avoids the complex channel estimation and guarantees the detection accuracy. Compared to the state-of-the-art coherent maximum likelihood sequence detection (MLSD) in the cases of static and time-varying channel response, the proposed OWNCD can gain similar to 1 dB and similar to 8 dB in signal-to-noise-ratio (SNR) at the 7% overhead forward-error-correction (FEC) limit (BER of 4.5 x 10(-3)), respectively, and can also reduce the computational complexity by 4 order of magnitude.