No nationalistic prejudice intended by the question. Just a debate on the possible outcomes of such a scenario which in my mind could be a very likely one. Again this is nothing to do with crazy leadership, but more to do with 'respond at all cost' to make one side take the other seriously.
A nuclear strike against a naval group as a last resort would not necessarily warrant a full out retaliation because the stakes then become considerably higher. What I'm getting at is can a limited conventional naval exchange actually occur between the super powers in these times given the current national rhetoric and assertiveness?
It's a good question, and best answered without emotion.
Reality check time. The scenario as constructed makes no logical sense. Again, absent bat-sh*t crazy leadership, a naval clash would not involve or result in the employment of nuclear weaponry, particularly against the US or US forces.
At this point, it really seems necessary for people to explain just what exactly they mean by a 'minor' naval clash. IMO most people would consider a 'minor' naval clash to range from naval vessels harassing vessels of another navy like some of the Iranian speedboats have sometimes done to US and allied naval vessels in the Persian Gulf, up to shots being fired between vessels using small arms and perhaps even ships cannon.
A 'major' naval clash would IMO be something more serious which results in a significant worsening of relations between nations, up to war. At the lower end of this would be something like the USS Stark incident, where an AShM was fired at a USN frigate by an Iraqi aircraft which caused serious damage to the US vessel and resulted in the death of over 30 crew. At the opposite end of the spectrum would be clashes which result in significant casualties and/or loss of a significant vessel, such events would usually be considered an act of war. An example of this (albeit the aftermath fell just short of the resumption of an active shooting war) would be the sinking of the ROKS Cheonan in March 2010.
The employment of nuclear weapons against the naval assets of another nation would absolutely be considered a warshot. Furthermore, when automatically assuming the dominance of a USN unit over that of a foreign navy, this is not discussing just individual ships, but flotillas, squadrons and fleets like a USN Carrier Battle Group (CBG) correct?
If that is true, then the use of a nuclear device against a CBG would almost certainly lead to the US responding with nuclear weaponry as well. Under the MAD doctrine the US would basically be forced to respond in kind. A failure to respond to such an attack with crippling if not annihilating force would only tempt and invite further nuclear attacks against US military and naval forces. The loss of a CBG would be significant, as a carrier itself has a crew of ~5,000 personnel and costs billions to construct, nevermind the cost of the embarked aircraft and munitions. Adding in the additional vessels which make up a CBG likely adds a further 1,200 - 1,800 personnel with a few billion more to construct the escorting cruisers, destroyers and frigates as well as any stores and replenishment vessels. In terms of power projection a CBG air group has more fighter/strike power than the entire air forces of most nations that are not amongst the Great Powers
So again, unless Russia or China was looking to get into a nuclear war with the US, the employment of a nuclear weapon against a US naval unit would not happen. Further, unless Russia or China was looking to get into a conventional war with the US, any direct clash with the USN would be minor and any damage inflicted on US vessels unlikely to require any important repairs.