The general form of a 2x2 unitary matrix is:
$$
\begin{pmatrix}
\alpha & \beta \\
-e^{i\phi}\beta^* & e^{i\phi}\alpha^*
\end{pmatrix},
$$
with the constraint that $|\alpha|^2 + |\beta|^2$ = 1.
Since you say that $U|0\rangle = U\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ b \end{pmatrix}$, we have that $\alpha = a$ and $-e^{i\phi}\beta^* = b$.
Therefore, the most we can say about the bottom-right corner is that $d=e^{i\phi}a^*$, and the most we can say about the top-right corner is $c=\beta = -b^* e^{i\phi}$.
So you have: $U|1\rangle = e^{i\phi}b^*|0\rangle - e^{i\phi}a^*|1\rangle$.
We therefore do not have enough information to determine the phase $\phi$, but since you only ask how to write $U|1\rangle$ we don't need $\phi$ because it enters our expression for $U|1\rangle$ only as a global phase.
In conclusion: If all we know is $U|0\rangle = a|0\rangle = b|1\rangle$, then we can say that $U|1\rangle = b^*|0\rangle -a^*|1\rangle$, which is correct up to a global phase.