Pearson correlation has been used for this purpose; see e.g. Exon level integration of proteomics and microarray data. However, it's generally used because it's the "default" correlation method that everyone knows, rather than because it's appropriate.

You should probably investigate canonical correlation analysis. There are implementations in many statistics packages; see e.g. CCP in R. CCA is commonly used in bioinformatics for multivariate correlation.

Another useful R package is DRI. It's designed for correlation of copy number and expression data, but I think it will give you useful ideas and perhaps could be adapted to use a matrix of values from proteomics.

One more R package: iCluster, for "integrative clustering of multiple genomic data types." Just requires 2 data matrices; one with your microarray data, the other with the proteomics data.