The channel nonlinearity and high-speed mobility in low-earth orbit (LEO) constellation communication are key bottlenecks that constrain the performance of waveform transmission. To address these challenges, we propose a universal continuous phase modulation (CPM) spread-spectrum communication method with high spectral efficiency, and a Doppler-insensitive signal detection mechanism. First, an enhanced dual-phase CPM spread-spectrum (DP-CPM-SS) waveform is investigated. By analyzing the principles and power spectrum characteristics of DP-CPM-SS, we correct the modulation index and design the optimalWiener filtering reception for CPM, improving the spectral efficiency and noise resilience. Subsequently, a CPM noncoherent detection integrating frequency estimation and phase pre-compensation is developed. The performance lower bound of the partial matched filter-fast fourier transform (PMF-FFT) frequency estimation algorithm is derived, revealing the relationship among the length and number of partial matched filters, the normalized frequency offset and the mean square error (MSE) of frequency estimation. Additionally, error probability performance and frequency offset adaptation range are analyzed. Numerical results show that the proposed method exhibits superior and stable performance under severe Doppler effects, which is a promising scheme for LEO constellation communication.