We report on the experiments on the interaction of gigawatt femtosecond laser pulses with suspended millimeter-sized water droplets. The transparent droplets experienced laser-induced breakdown and explosive boiling up and emitted a broadband radiation. This radiation covers the spectral range from 450 to 1100 nm and consists of the spectrum of laser pulse scattered and transformed by the droplet due to self-phase modulation and plasma emission produced in water during photoionization. The droplet emission spectrum showed remarkable broadening at all viewing angles and is maximal in the direction of the laser exit from the droplet. The enlargement of the droplet results in additional spectral spreading of the emitted radiation. The depth and amount of laser pulse spectral self-transformations upon propagation through the water droplet are simulated by means of numerical calculations.