It could happen if the CCP waits long enough to suppress the protests. IOTL they had already spread to other cities; a week or more could be decisive. I'm not well-versed on the details of how the Politburo made the decision to crack down, but I understand there were people who were for and against (the against side got purged). If the CCP could be paralyzed for even a bit longer, you could have the hardliner faction resort to military force without consensus, and thus create splits in the PLA itself. IOTL it was reported that some generals refused to turn on the people, and had to be replaced. In this scenario it could really become fragmented.
As for the effects of a civil war, my impression is that it would be relatively low-intensity, except perhaps in places with large numbers of ethnic minorities. Communism is only still compelling as an ideology IOTL because of the stability the CCP brings; if a war has broken out I don't think that people and soldiers would fight desperately to support its political doctrine. We would probably see some kind of military junta emerge victorious after a short flurry of confrontations at strongholds around the country; in this sense it would appear similar to the events of 1911 where General Yuan Shikai jumped on teh anti-Qing bandwagon and placed himself in the presidency. Unlike the Warlord era, China in 1990 wouldn't be fragmented due to the strong organizational tradition of the CCP and the strength of Han Chinese identity. The greater harm would be that done to the economic progress made during the 1980s—it would set China back ten or twenty years, which could be crippling.