Once the states seceeded, the Constitution is irrelevant.
Lincoln could either see the states as violating their Constitutionional obligations -- thus being illegitimate governments deserving of being crushed, or Lincoln could have viewed the break-aways as foreign powers hostile to the remaining unions states (as the secessionist states proved by seizing federal forts and attacking Fort Sumter.)
So either way, illegitimate state governments or foreign hostile powers attacking US interests -- Lincoln and the US congress had full legal and moral rights to defend US interests.