No, the uncertainty principle is a statement about the localization of a quantum state when represented in position vs. momentum space (more generally, the eigenbases of two non-commuting observables). So long as no measurements take place, the quantum state can be evolved completely deterministically forward and backward in time if you know the Hamiltonian. This is what physicists mean when they say quantum mechanics is deterministic.
Any state, even one translated forward or backward in time using the Hamiltonian, will obey the uncertainty principle: namely the spread of the state in position space must obey a certain inequality in union with the spread of the state in momentum space. This does have some net effect on the outcome of measurements of that quantum state, but that is not the same thing as the state itself.
The state itself evolves determanistically forward or backward until a measurement occurs. Of course, measurements on the state like "what's your position" or "what's your momentum" will also obey the uncertainty principle, because the outcome of measurements (the part of quantum mechanics that is random) are probabilistically determined by the amplitude of the wavefunction over the eigenspace of the measurement operator. That is to say, the uncertainty principle is not related to randomness of quantum mechanics in any way other than requiring that the state will always have some spread over the eigenbases of non-commuting operators.