To achieve high quality lighting and visible light communication (VLC) simultaneously, GaN based white light emitting diodes (WLEDs) oriented for lighting in VLC has attracted great interest. However, the overall bandwidth of conventional phosphor converted WLEDs is limited by the long lifetime of phosphor, the slow Stokes transfer process, the resistance-capacitance (RC) time delay, and the quantum-confined Stark effect (QCSE). Here by adopting a self-assembled InGaN quantum dots (QDs) structure, we have fabricated phosphor-free single chip WLEDs with tunable correlated color temperature (CCT, from 1600 K to 6000 K), a broadband spectrum, a moderate color rendering index (CRI) of 75, and a significantly improved modulation bandwidth (maximum of 150 MHz) at a low current density of $72\mathrm{A}/{\mathrm{cm}}^2$. The broadband spectrum and high modulation bandwidth are ascribed to the capture of carriers by different localized states of InGaN QDs with alleviative QCSE as compared to the traditional InGaN/GaN quantum well (QW) structures. We believe the approach reported in this work will find its potential application in GaN WLEDs and advance the development of semiconductor lighting-communication integration.