Will another American Civil War erupt soon?  

Actually this bill makes it a violation of state law to be in the state of Arizona if one is not there legaly. It is arguably unconstitutional as immigration is an enumerated power to the federal government. Arizona could how ever argue that it is protecting it's state borders and is well with in it's rights to do so. They could also deploy their own state military to the border if they chose to.
There are a certain number of powers given to the federal government in the constitution, anything outside those limits is reserved for the states and people, as per the 9th and 10th amendments to the U.S. Constitution. Therefor, most things do in fact fall to the state and local governments. the federal government regularly steps over it's bounds in many areas that are state authority, many states are now just starting to push the federal government back to it's constitutional limits.
This next part is extremely important....the Militia is not a "movement", it is an institution. Do not buy in to the media hype that the militia is a domestic terrorist movement. The militia is a lawfull defense force, contstitutionaly defined in most state constitutions. I don't know what state you live in, but look up your state's constitution and odds are it states right in it that YOU are in fact considered Militia. In many cases it is simply defined as any able bodied citizen able to carry arms in defense of the state and the constitution, against any enemy. In fact, the primary enemy the Militia was established to defend against was a tyranical government, even if that tyrany were to ever come from our very own government..
"What, Sir, is the use of a militia? It is to prevent the establishment of a standing army, the bane of liberty. ...Whenever Governments mean to invade the rights and liberties of the people, they always attempt to destroy the militia, in order to raise an army upon their ruins." ~Elbridge Gerry
I think we are heading for another civil war of sorts. It is hard to imagine what shape that will take. It will not likely be as the first one. I envision it more as the states standing up and putting the federal government back in to it's rightfull constitutional roll, through reassertions of their 10th amendment rights as soviergn states. There are a couple states that have gone as far as to pass general laws that make the enforcement of unconstitutional federal laws a felony in those states, opening the door to arrests of federal agents caught with in those state attempting to enforce said unconsitutional federal laws. I think the next civil war will begin as a law enforcement action by numerous states with an eventual constitutional convention convened by the states to nullify the federal government and re-establsish it.

There has been the risk of another Civil War for some time now. The Federal Government has repeatedly denied States even the most basic rights or powers that Constitutionally should belong to the state. It has been a powder keg waiting to go off and when the Federal Government pushes once to often they are going to find themselves in a war on multiple fronts. Alaska already has a group pushing hard for secession from the United States, the Southern (also called Confederate States) have been biding their time until they could gain their freedom, Texas has been biting at the bit and California is expected by many to be the state that is going to lead the way in the bid to end the corruption of the Federal Government.

As for Arizona's new anti-illegal immigrant laws. Good for them! Seriously the Federal Government refuses to pick up illegals who have been arrested, what else is a state supposed to do save for taking matters into their own hands.