Label-free detection of DNA molecules on chemically vapor-deposited diamond surfaces is achieved with spectroscopic ellipsometry in the infrared and vacuum ultraviolet range. This nondestructive method has the potential to yield information on the average orientation of single as well as double-stranded DNA molecules, without restricting the strand length to the persistence length. The orientational analysis based on electronic excitations in combination with information from layer thicknesses provides a deeper understanding of biological layers on diamond. The pi-pi* transition dipole moments, corresponding to a transition at 4.74 eV, originate from the individual bases. They are in a plane perpendicular to the DNA backbone with an associated n-pi* transition at 4.47 eV. For 8-36 bases of single- and double-stranded DNA covalently attached to ultra-nanocrystalline diamond, the ratio between in- and out-of-plane components in the best fit simulations to the ellipsometric spectra yields an average tilt angle of the DNA backbone with respect to the surface plane ranging from 45 degrees to 52 degrees . We comment on the physical meaning of the calculated tilt angles. Additional information is gathered from atomic force microscopy, fluorescence imaging, and wetting experiments. The results reported here are of value in understanding and optimizing the performance of the electronic readout of a diamond-based label-free DNA hybridization sensor.