I expect negative but I’ve never tried that. This is, because both bacteria grow very different. On a Columbia agar, you can discriminate both by their growth characteristics. While S. aureus typically grows in creamy, yellow, sometimes a bit grayish but slightly arched colonies, surrounded by a clear beta-hemolysis, Pseudomonas grows in grey, flat colononies which have a metallic luster (and usually no hemolysis). Moreover, P. aeruginosa smells intensively like lime-tree blossom (a sweet odor). If you do a darkfield microscopy, you see cocci (S. aureus) or rods (Pseudomomas). The coagulase test is intended to be used in case you want to discriminate between S. aureus (positive) and so-called coagulase-negative Staphylococci (mostly, but not always negative).

Both the organisms are very different, you can check with biochemical reaction for P. aeruginosa.  Simple gram's method of staining will be sufficient to rule out Staphylococcus. By IMViC test also you can differentiate.

Thanks a lot for your answers

The right way forward in bacterial identification (not in labs where automated methods are employed)  is first look at growth and colony morphology in conventional and special media and also examine a gram stained smear. There are standard algorithms for further biochemical characterization based on the grams reaction and growth on media (like LF on MacConkey etc.). if one follows that strategy, a situation to consider coagulase test for a GNB will not arise.

Regarding you query, yes false positive coagulase test is possible with Pseudomonas. That is usually associated with use of citrated plasma.  Pseudomonas and such citrate utilizing bacteria can utilize citrate and  cause false positive coagulation! This is usually associated with slide format of the test and therefore the tube coagulase is what is recommended.

Hope this was useful.

Coagulase test is used to differentiate coagulase-positive S. aureus from coagulase-negative staphylococci. Pseudomonas aeruginosa gives negative result for coagulase test because it doesn't produce coagulase enzyme.