Kirk was right to offer assistance and, in fact, he did not need Nero's permission to provide assistance and what he was effectively trying to do was to take Nero into custody. This is absolutely the Federation way. Apart from that, the ship contains important technology, details about Klingon defences, etc. There may also be prisoners on board. Saving the ship is high on a list of priorities, although not as high as bringing Nero to justice.
However, what Kirk should not do is endanger his own ship while trying to efect an arrest/rescue.
In staying as close as he does in order to actively destroy the ship, he is carrying out a summary execution of everybody on the ship without an individual assessment of their innocence or guilt of any particular crimes. That is definitely not the Federation way and conjures echoes of indiscriminate attacks by drones on legitimate targets that kill civilian bystanders i.e. Kirk has no knowledge of innocent prisoners on the ship and therefore assumes that there are none. He also endangers his own ship and crew. That is also not the Federation way.
There are passable reasons for destroying the ship - if it travels through time again, the technology could cause havoc but dialogue suggests that the singularity will destroy it. The singularity will also prevent the ship from firing weapons. It's not clear how comms still work either but the fact that they do could mean that people could have been beamed off the ship and, given the range of the transporter established earlier in the same film, this could have been attempted from a much safer distance (or while travelling to amuch safer distance).
To summarise, the scene was very sloppily handled. Kirk doesn't explore his most obvious and safest options, and executes helpless prisoners. Not the Federation's finest hour.