Is there any unitary transformation that has the effect of only multiplying the diagonal with some values.
For example if I start with the matrix
$$A=\left(\matrix{1&a&b&c\\d&1&e&f\\g&h&1&i\\j&k&l&1}\right)$$
Is there some unitary matrix which will transform it to
$$A=\left(\matrix{\alpha&a&b&c\\d&\beta&e&f\\g&h&\gamma&i\\j&k&l&\delta}\right)$$
where we have some target $\{\alpha,\beta,\gamma,\delta\}$.
Suppose $A = I$, the identity matrix. Then your condition says that $U^* U$ is equal to some non-identity matrix, contradicting the definition of a unitary matrix.