Overall, calling it the revenge for 1600 would be a stretch, but there were certainly some Satcho participants who perceived it in those terms.
There were longstanding historical tensions between the Tokugawa shogunate and Satsuma and Choshu domains respectively. Satsuma and Choshu were losers at the Battle of Sekigahara in 1600 against Tokugawa Ieyasu's victorious side, and Choshu lost a great deal of territory as a result. These historical tensions contributed to their deteriorating relationships during the Bakumatsu (end of the Shogunate) era. And both Satsuma and Choshu had romanticized traditions that one day they would regain independence (and more power/influence over their former lands in Kyushu and Chugoku) vis a vis the Tokugawa. These domain grudges were particularly emphasized by Satsuma and Choshu writers in the Meiji era. So you may be familiar with some of their most famous traditions. The following references via Albert Craig in Choshu in the Meiji Restoration are often cited in accounts of the Meiji Restoration.
IMPORTANT EDIT: See this discussion casting doubt on the accuracy of the following accounts.
Most of this anti-Tokugawa bias was formless, a part of the Choshu heritage, but in a few cases it seems to have crystalized into specific practices. One ceremony embodying this animus was held annually on the first day of the new year. Early in the morning when the first cock crowed, the Elders and Direct Inspectors would go to the daimyo and ask, "Has the time come to begin the subjugation of the Bakufu?" The daimyo would then reply, "It is still too early; the time is not yet come." While obviously secret, this ceremony was considered one of the most important rituals of the han. Another comparable custom in a more domestic setting has also been recorded. Mothers in Choshu would have their boys sleep with their feet to the east, a form of insult to the Bakufu, and tell them "never to forget the defeat at Sekigahara even in their dreams." In the case of Satsuma, every year on the fourteenth day of the ninth month the castle town samurai would don their armor and go to Myoenji, a temple near Kagoshima, to meditate on the battle of Sekigahara. And on the following day, they would return to the castle town to listen to "The Military Record of the Battle of Sekigahara." (Sekigahara gunki.) (p. 21-22)
But these grudges, though they came to prominence when their relationship with the shogunate was souring, weren't the driving motivations of the Meiji Restoration for Satsuma or Choshu. Both domains had issues with the shogunate but maintained entirely functional relationships with the shogunate during much of the Edo Period. Satsuma in particular provided two primary wives to Tokugawa shoguns in the 19th century, and achieved some influence in national circles in the late Edo Period. Choshu always was further away from the shogunate, but even through the tumultuous late 50s to mid 1860s there were factions in Choshu working to stabilize the domain's relationship to the shogunate. It took a lot to make a complete revolution against the Tokugawa even thinkable. Even the most die-hard Imperial loyalists in 1864, the year of Choshu's first battle with the shogunate (and Satsuma), hardly imagined overthrowing the shogunate, just reducing the Tokugawa power and making them collaborate with the other great domains. This vision of the Great Lords all working together (with each great domain ambitiously imagining their own proper influence in this set-up) had to die before a full destruction of the shogunate was on the table.
How that vision died involves
And that's not an exhaustive list of everything that contributed to this mess. The Bakumatsu was a rollercoaster ride where nobody knew where they were going until they got to the very end, and revenge was never the big motivating factor. Yet those old grudges certainly greased the wheels of this metaphorical rollercoaster.
For your interest: My first replies on AskHistorians include further elaboration of Satsuma's historical issues with the shogunate, including an explanation of how the post-Sengoku order gave Satsuma and Choshu an overproportion of samurai in their populations.
Can I say as postscript that Choshu's particular history in regards to Segikahara is really, really weird? As in, the Mori seem to have tried to play both sides of the conflict back in 1600. Daimyo Mori Terumoto, the commander-in-name of the Western forces didn't even show up on the battlefield with his forces, and his cousin Kikkawa Hiroie cast his lot in with Tokugawa Ieyasu, stopping the bulk of the Mori troops on the scene from actually fighting. There's a lot of speculation Terumoto was hedging his bets as the wind blew, though Kikkawa Hiroie could also have been moving on his own.
The Tokugawa expertly exploited the situation by making the Kikkawa daimyo in their own right, and giving them some of the Mori territory. The Mori in return refused to recognize the Kikkawa as anything but their retainers holding Mori land, while the Kikkawa prudently tried to stay deferent to both the Mori and Tokugawa: a pretty impossible position. The Mori and Kikkawa only reconciled in 1863.
So, the Mori didn't actually have some noble Sekigahara defeat to harken back to, but a weird in-clan . . . something. Did that stop 1860s Choshu samurai from idealizing the Mori's valiant last stand at Sekigahara? No it didn't. God, I love Choshu domain.