Semicolons replace periods (end stops, the "dot" you refer to), so it doesn't matter grammatically because in both cases, the sentences are grammatical.
The syntax and semantics, however, need work. The last clause is out of order, or else the punctuation has to be changed, as well as some other things.
I'd rewrite this sentence as:
A. Alex feels pain in his leg and shouts. Then he rubs the sore spot with his left hand and looks at the leg. His leg has swollen a bit.
B. Alex feels pain in his leg and shouts. Then he rubs the sore spot with his left hand and looks at the slightly swollen leg.
First he feels the pain; then he shouts: the pain prompted the shout, not the other way around. The final clause has to be in present perfect, it seems to me, because the story is being told in the immediate present rather than the past tense.
C. Alex feels pain in his leg and shouts. His leg swells a bit. He rubs the sore spot with his left hand as he looks at the leg.
I'm sure that other users could come up with better sentences, but the point is that there's more than one way to write this description. Another point is that there's no necessity to use a semicolon to connect the sentence containing the source of the pain in his leg (hitting it against a rock or tearing a tendon) with the sentence that contains the consequence of his injury. The connection is semantically obvious and would be so if told orally rather than in writing.
EDIT: There are numerous websites that explain semicolon usage, e.g., this one. Here's one answer: "Rule 1: Use a semicolon between independent clauses that are closely related in theme." But this means: You may use a semicolon. It doesn't mean You must use a semicolon. Using a semicolon is a style choice, as Edwin Ashworth's comment says.
I frequently use semicolons for biomedical papers. I use them to replace conjunctions and transitions because I want to show a close connection between two independent clauses without using yet another conjunction in the sentence. It's formal punctuation and generally unnecessary for informal writing. It's rather unusual punctuation when writing stories. In this example: "Bob kicked Tom hard in the groin; Tom staggered and fell to the ground." versus "Bob kicked Tom hard in the groin. Tom staggered and fell to the ground." using the semicolon is grammatically correct, but nothing seems to be gained by using it or lost by using an end stop. The cause-effect relationship is clear from the content, not the punctuation. Another, from the linked website: "Jane visited her parents; they had to take their dog to the vet." Using the semicolon to replace "because" seems wrong to me because the only thematic connection is Jane's parents. That makes the second clause seem non sequitur because it clashes with the choice of "visited". If, indeed, her "visit" was to help them because they had to take the dog to the vet's office, the sentence needs to be recast and the conjunction because needs to be there for clarity.
Novelists often use commas instead of end stops to create what some of us call "comma splices" to try to make their prose more fluid, e.g., "Bob kicked Tom hard in the groin, Tom staggered and fell to the ground." I'm not a fan of this kind of "creative writing".
I'd also advise against using semicolons when telling stories. Most readers don't like them. Hemingway's punchy prose changed the way a lot of novelists wrote. Thomas Mann, however, loved semicolons, but he wrote in German. Kurt Vonnegut said this: "'Here is a lesson in creative writing. First rule: Do not use semicolons. They are transvestite hermaphrodites representing absolutely nothing. All they do is show you've been to college.'
― Kurt Vonnegut, A Man Without a Country" He could have used a semicolon between nothing and all had he wanted to be ironic. I agree with him: save semicolons for formal expository prose. Grammaticality is not a sufficient criterion for creative writing. Creative writing is all about style, plot, characters, sales, and what pleases your readers. Semicolons are generally not pleasing.