In HyperLedger Fabric, to get details of a given transaction, one can use QSCC system chaincode call:

GetTransactionByID

The problem I came across, was with parsing the response. I couldn't find any good explanation how to do this.

Finally, with a bit of experimentation, I've managed to figure it out. Sample code and description are available here:
https://github.com/msedzins/GetTransactionByID