If you throw in the residue degrees as well, then you get Zev's question. Otherwise, the converse is not true. As an example, consider a quadratic field that has an unramified everywhere $A_5$ Galois extension (see e.g. [this MO thread][1]). Since $A_5$ is simple, any proper intermediate extension will not be Galois over the quadratic. But the ramification indices will all be 1, since they are all 1 in the big extension.


  [1]: https://mathoverflow.net/questions/44801/a-5-extension-of-number-fields-unramified-everywhere