@paninid p-values, to a large extent, exist because calculating the posterior is computationally expensive. Not all fields use the .05 cutoff.
A p-value is an #estimate of p(Data | Null Hypothesis). If the two #hypotheses are equally likely and they are mutually exclusive and they are closed over the #hypothesis space, then this is the same as p(Hypothesis | Data).
Meaning, under certain assumption, the p-value does represent the actually probability of being wrong.
However, given modern computers, there is no reason that #Bayesian odds-ratios can't completely replace their usage and avoid the many many problems with p-values.