I don't think so. Nucleic acid contamination would consist of molecules having a wide range of sizes, so no bands would form. Large molecules would not enter the gel. Smaller molecules, being very strongly negatively charged, would run off the gel. Even if the nucleic acids were of a single size that remained on the gel, which is unlikely, they would not stain with Coomassie Blue.

SDS-PAGE is specific for visualizing proteins present in the sample. The fundamental principle of Coomassie Blue staining involves the interaction between the negative sulfonic acid groups of the dye and the positive amine groups in the protein. It is a general stain that stains all proteins. DNA and RNA being nucleic acids will not be stained and hence any nucleic acid contamination in your sample will not be visible on your SDS-PAGE gel.

It is possible to run PAGE gels for DNA but it's a different process and doesn't involve SDS or Commassie staining. So even though you could run DNA samples and visualise those you most certainly can't do both at once. The only way DNA contamination might be visible is indirectly where e.g. running protein samples with high amount of DNA in (especially genomic) is going to interfere with the migration of your proteins (this is also usually visible during loading of the samples as the denaturing loading buffer breaks apart the DNA and sample becomes viscous).

Can DNA -pigment conjugate of 15 kd be resolved on native or sds page?