correlation value of 0.6 is moderate value. The square of correlation value is 0.36 which means 36% of the dependent variable can be explained by the independent variable. 

correlation value of 0.6 is moderate value. The square of correlation value is 0.36 which means 36% of the dependent variable can be explained by the independent variable. 

The correlation coefficient (CC) translates a trend, which can be replaced by a regression curve or not, according to the scatter plot concentration: CC = 0.9 allows very good fits; CC = 0.8 allows good fits; CC = 0,7 is acceptable; CC = 0.6 would be admissible in some cases, and CC = 0.5 or lower should be usually rejectable. You can calculate the 95% confidence interval of CC = 0 also, to see how wide or narrow is (the lower the simple size the higher the probability of null correlation).

The field "in medical research" is too wide to give any useful answer. There are applications/experiments where 0.6 would be extremely good, and there are applications/experiments where 0.6 would be extremely bad. I don't think that it is helfpul to interpret or rate a correlation coefficient without being very specific about the experiment and its context.

It would be possible. However, considering the CC range from 0 to 1, positive or negative, a CC lower tan 0.5-0.6 seems to be not extremely good.

The CC is a measure combining the strength of the dependency (call it "effect size") and noise. In correlation analysis this is aditionally complicated by the fact that ratio of effect and noise depends on the span of values that is measured. In some experimetal settings there is neccesarily a lot of noise, and seeing a CC of 0.5 indicates a rather strong effect.

Then you are talking about partial correlation

Thank you all for the explanation. To be more specific, I found correlation coefficient of 0.6 between platelet recovery and platelet storage days. So, I would like to know whether this can be a significant in medical research as in medical science lot of factors affects the platelet recovery in patients. Yes I agree that 0.6 may not be good strength of relationship but in medical research whether this can be considered as good strength considering the factors affecting it?

A significant inverse correlation between this 2 variables has been reported in several studies. The suspensión in plasma or additive solution (PAS), the composition of PAS, storage temperature, time from collection to storage, etc., are relevant factors affecting the PLT recovery. r = 0.6 might be considered good, but not very good, considering the impact from other factors 

Correlation Analysis. Achim Buyul and Peter Tsefel’s classification was employed to estimate correlation coefficients (r): up to 0.2 - very weak, up to 0.5 - weak, up to 0.7 - medium, up to 0.9 is high and over 0.9 is very high correlation

Achim Bühl, Peter Zöfel. SPSS Version 10. Einführung in die modern Datenanalyse unter Windows, 7, überarbeitete und erweiterte Auflage, Diasoft: 2005, 602 p.