I doubt the white spider is the culprit. The likely answer is that you've got something else making or causing the slashes and the spider just crawled into one of them.

Could it be that your fruits are splitting or cracking?

Spiders don't eat plants aor parts thereof.

Most likely the fruits are cracking on their own. The remedy? Harvest daily.

SPIDERS EATING TOMATOES!!!!!!!!!!

Not a chance the little feller eats things that do eat maters, "spiders are great.
I do know that some are bad in the wrong environment but I can only think of two.
Black widow and Brown recluse and your not likely to find them in the garden.
Folks I let these little guys crawl on me and have since I was 9 or 10.
I am as passionate about spiders as Tomato Worm 59 is about hornworms.
I just love the things.

Worth

I have the same thing that happens, but it is usually because the fruit is old. They either split down the side on the vine or after picking and letting them sit for a while.

I have grown tomatoes for several years now. I started noticing small pecking holes in my tomatoes this year, large and cherry. The holes are only on the ripe, tasty tomatoes. ( of course! ) I was chalking it up to the blasted birds. Then as I went to snap a ripe, lucious tomato off it's stem, a bunch of little brown spiders with red butts (from pecking on my ripe tomato) ran in all directions from the hole they had eaten into my beautiful tomato. Yes, there are tomato eating spiders. Blast it!

Doubtful. There is only one known spider species that eats plant matter, and it has a very specific diet that does not include tomatoes. You were almost certainly seeing the nymphs of stinkbugs or squash bugs or related bugs, which often look like little spiders with red or brown bodies. And the "pecking" was probably still from birds or mice or something, and smaller pests were just being opportunistic and taking advantage of broken skin or just coincidentally there. The pecks could even be from birds trying to eat the bugs and coincidentally stabbing your tomatoes as well, although it's far more likely to be birds eating your tomatoes and the bug nymphs coincidentally hatching/feeding in the vicinity. Maybe, though very unlikely, the spiders were newly hatched and thirsty and drinking the liquid from the tomato. But newly hatched spiders are extremely tiny, and spiders only congregate in groups when they are first hatched, as almost all species will eat one another if they hang out together in groups. Bottom line: spiders will not eat your tomatoes and are basically always beneficial in the garden.

Twice this week I left ripe cherry tomatoes on my kitchen counter and in the morning found little bites taken out of one of them and bits of skin on the counter. The second time, there was a large brown spider in my sink. I’m pretty sure the spider was eating the tomato.

No, spiders do not eat tomatoes. And if they did, they wouldn't leave bite marks. They may eat pests eating your tomatoes, though.

Personally, I suspect you have a mouse.