If a batter hits a walk-off winning home run in the bottom of the 9th inning, the umpire signals it a home run, but the batter fails to step on all four bases while rounding them in his home run trot that won the game, is the defensive team allowed an appeal? Can a defensive player, since the umpire signaled home run ending the game, request another ball be put in play, after the game ended, request the umpire call Play Ball as required under baseball rules, even though the game is already over, and upon hearing this - can the defensive player then touch the runner, throw to the base or touch the base on appeal with a new ball, thus putting the home run hitter out on Appeal, and eliminating the home run, thus continuing the game? I have had trouble locating an answer to this one for years…Thanx…
…and, to finish the story, the teams did end up in a tie for first at the end of the 1908 season. The Cubs won a single-game playoff, and went on to beat the Tigers in the World Series. And they haven’t won since. The Curse of Merkle!
Of course, in the Merkle case, the winning run scored on a single; if it was a home run, no force out would have been possible. I Googled a bit but couldn’t find a rule on this home run thing. What if you hit a walkoff solo home run and twisted your ankle rounding second? Would they make you crawl the last 180 feet?
An interesting case is Robin Ventura’s “grand slam single”, in which Ventura hit a game-winning grand slam but was only credited with a single and one RBI because he was mobbed by his teammates and only touched first base.
History aside – yes, the defending team can call for a baseball and run an appeal play. The game is not over until the runner legally touches home; if he mises a base on the way, the run does not count (assuming it’s a solo home run to win the game; it gets more complicated with men on base*) if the play is appealed and the defending team gets the ball to the base before the better has touched it. I would assume the team can ask the umpire to throw them a ball instead of having to retrieve the home run ball.
That would apply even if the home run is in the first inning: the defending team would call for a ball and touch the base to put the batter out.
(*if the winning run is scored before the batter reaches the base he forgot to touch, the run counts and it’s a moot point.)
Just slightly off-topic, in last night’s Cardinals-Astros game, pinch hitter Roy Oswalt (normally a pitcher) hit a double in the 11th inning. The Cardinals appealed, saying Oswalt had missed touching first base, and Oswalt was called out to end the inning.
Oddly enough, this was a central plot point in the 1964 novel A Pennant for the Kremlin. The batter had hit a home run, but tripped over third base and fell, unable even to crawl. His teammates picked him up and carried him to home plate. However, one of the baserunners crossed behind the batter to help carry him, and the batter was called out for passing the runner.
No, you can be replaced by a pinch-runner. A similar event happened last September 14 (the home run wasn’t walk-off, but it makes no difference), when Gabe Kapler injured himself while running the bases on Tony Graffanino’s home run. He was replaced by a pinch-runner who completed his circuit of the bases, ahead of Graffanino.
The answer is, yes, there can be an appeal after a walk-off home run.
When Bobby Thomson hit his famous home run to clinch the pennant for the Giants in 1951, Jackie Robinson watched the runners to see whether they missed any bases. Only when Thomson touched home plate did Robinson turn and walk off the field.
What a haven baseball is for pedants and rules-lawyers! Cricketers would find it weird if a batsman, upon hitting a boundary, had to physically walk or run the runs, still more if someone could be run out on a technicality while doing so. We just rack up the runs (four or six) and get on with the game. 
I think the question has been answered already, but it is possible for home runs to be appealed after weeks have passed. Recall the infamous pine tar incident with George Brett and the Royals vs. the Yankees. Brett was called out after homering for having too much tar on his bat. The Royals appealed this to the league office after the game was over. The league upheld the Royals, awarded the home run, and ordered that the game be replayed from that point on. So on the Royals’ next visit to New York they picked up the game from that point. Before the first pitch following Brett’s homer, Billy Martin comes out of the Yankee dugout and claims that Brett hadn’t touched all the bases and that these umpires couldn’t prove him wrong since they weren’t there for the first part of the game. The umpire calmly pulls out a notarized statement from the umpires that were there that confirmed that Brett did indeed touch all the bases weeks earlier. Yankees lost that appeal and the game.
And baseball players would find it weird if a batter hit a ball and then decided he didn’t want to run and instead wanted to take another at-bat.
Given that cricket matches often last longer than the lives of some of the participants, I’d say cutting a few minutes here and there is a bit more important in that sport.
Well quite, but if we wanted to have to run every time we hit the ball we’d play “tip and run” which is considered more suitable for children. 
Also, we find it odd that a batter can be out when the pitcher hit nothing but fresh air.
That hardly ever happens (see Abdul Aziz). 
Since we are getting pedantic with the rules, the umpire is only required to call, “Play.” (Rule 4.02 & Rule 5.01)
Why? If a batter was allowed to sit there and take pitches constantly, the leadoff hitter would just sit there and take dozens of pitches to wear the starter out. If it’s a hittable pitch, the batter should at least make an attempt to swing.
Technically, he’s got to hit the catcher’s mitt on the third strike.
Technically, a catcher’s so-called “mitt” is properly called a pud.
And the pitch doesn’t have to hit the catcher’s pud on the third strike, but only be caught before the pitched ball hits the ground, and the catcher can catch it with anything he wants to.
The pitched ball could become wedged between the bars of his mask, for instance, whereupon he could remove it with his hand to show the umpire that he had control of the pitch, thus getting the required put out.
There … out pedant that!
(I’ll bet 30 Quatloos that someone does!)
Actually, according to rule 1.12, it’s a mitt.
And if the ball lodges in the catcher’s mask on strike 3 and remains out of play, the batter is entitled to first base.
And no, he can’t catch it “with anything he wants to.” Catching the ball with any part of his uniform other than his mitt would be illegal under the definition of a “catch” in rule 2.
I guess technically according to the definition of “catch,” the catcher could bare-hand it, so my original statement should be amended.
This is as good a time as any to use the phrase “a whole 'nother ball game”, I guess. In cricket, nearly all sides will go into the game with at least four bowlers, often five or more, partly so as to rotate the bowling duties (which they must do every six fair balls in any case) and partly to present the batsmen with a variety of challenges.
Standing there and taking lots of deliveries to take some of the sting out of the opening bowlers is indeed part of the technique of an opening batsman. There’s a balance, though; it’s also a strategic advantage to score runs when they’re there to be scored - particularly since, at the beginning of an innings, the field placings will be quite aggressive (cricket fielders have considerable liberty as to where they stand) while the bowlers are fresh and the ball is at its bounciest, and a batsman prepared to take risks can advance the score quickly.
Since the batsman must, at minimum, defend a physical target that is larger than his bat, he must at least play at the ball if it is bowled straight, but the bowler is not rewarded simply for beating the bat if he doesn’t also hit something.