Not at all. First and foremost the Slesvig-Holsteiners that rebelled against the crown seem to have considered themselves Germans not Danish and in fact wanted a Slesvig-Holstein state that would join the German Confederation as they considered it better to belong to a larger more powerful entity in accordance with the "threshold theory." Am not sure what the proper English term should am translating it from the book am reading from, but it was an idea popular in the 1800s that smaller states were doomed to unite into larger entities or be devoured by their more powerful neighbours. Many of the national union movements (e.g. the Italian and German ones) were influenced by this type of thinking.

Secondly, it was the German Confederation's army, mostly supplied by Prussian troops that ensured the rebellion wasn't utterly crushed by the Danish national army. One of the reasons Prussia couldn't push the matter fully and the German Confederation had to reign the Slesvig-Holstein forces was because Russia as a guarantor for Danish suzerainty was amassing troops on Prussias eastern border threatening intervention. It also featured participation by the Swedish-Norwegian army (as peace keepers) and effectively protecting some of the Danish islands from Prussian invasion. There was also some unofficial volunteers from Sweden-Norway fighting on the Danish side as the matter was in many circles considered a Scandinavian fight and not solely a Danish matter. Again due to the threshold theory it was seen that the Scandinavian countries needed to unit or they would be devoured by Prussia and Russia respectively. If Denmark lost it would become a Prussian fief. If that happened Sweden-Norway would likely be the target for Russian expansion was the view. Hence, to protect Scandinavia Denmark had to be defended at the river Ejder.

To make the matter more complicated there were political and dynastic issues clouding the waters. The Danish royal line had no heirs beyond the extant crown-prince who wasn't expected to provide any issue so the eventual next in line for the throne were the cadet branch that was duke of Slesvig-Holstein-Lauenburg was expecting to inherit the Danish crown. Holstein was part of the German Confederation while Slesvig was not. The duchies had been in personal union with Denmark since medieaval times (except Lauenburg). The dying dynasty was also scheming with the Swedish-Norwegian dynasty to try and create a Scandinavian untied kingdom. Other politicians were trying to use the issue to topple the kings, prop up absolutist rule, create a Scandinavian union, create a Danish nation state by getting rid of the German duchies (but keeping Slesvig), keep a Danish "wholestate" and so on. Sometimes all or several at the same time.

The spark for the war was revolution in Denmark that toppled the absolute monarchy and ushered a more liberal constitution. The Duchies refused to accept this constitution creating another faultline between "liberal" and "conservative".

In other words, this is highly complicated and not without reason did Lord Palmerstone say:

The Schleswig-Holstein question is so complicated, only three men in Europe have ever understood it. One was Prince Albert, who is dead. The second was a German professor who became mad. I am the third and I have forgotten all about it.

With so many outside players, all great powers had some different agendas they tried using the Slesvig-Holstein question to push it's not really right to call it a civil war.

I have really only touched the surface of this using Rasmus Glenthöj and Morten Nordhagen Ottosen "Union eller undergång" parts 1 & 2 (2022) [Swedish translstion]. Their book really covers the Scandinavianism of the latter 1800s but the Slesvig question features heavily in it as it is one of the major problems for Denmark. Half it's nation is part of a German sphere of influence on the verge of uniting. And the Slesvig question becomes a pawn used in the various political manoeuvrers, both domestically and internationally.