Does going vegan roughly save 600 gallons of water a day?

Short answer: No.

It's possible to save as much, but probably not average. It depends on what the vegan eats. However, it'd be fair to say an average vegan diet saves about 300-500 gallons of water per day.

Initial investigation

The source of the quote links to National Geographic, "Water Conservation Tips". This article says:

On average, a vegan [...] may indirectly consume less than a person who eats dairy, though a new study calls this into question [...]

It mentions the 1000 gallons per day figure for the Standard American Diet, but I couldn't find the 400 one for vegans.

The "new study" appears to be this. The key point in that study is that while in terms of water/kg, non-meat products are generally more efficient, in terms of calorie/kg, they're not all as such. So, does "going vegan" save 600 gallons of water per day? Not if the vegan starts only eating lettuce, but maybe if they cut all the meat products from the American diet.

What follows are some calculations. I took figures from here and a related PDF document.

We'll see figures for the consumption of dairy and meat, which is where vegan and non-vegan diets mostly differ.

Comparisons

Vegan vs. Standard American Diet

The Standard American Diet consists of roughly 50 pounds (22.7 kg) of beef, pork, and chicken per year (and other non-vegan stuff). Dairy is about 40% of the recommended, intake, so about 2 cups, which'd cost about 500 litres of water.

This costs about 2110 litre of water per day (((4325+5988+15415) * 22.7)/365+510) (assuming 1kg/L for milk). Or about 557 gallons. Add water footprint for various other meat-related products, and ... maybe that's where the figure comes from? A vegan just cuts all that and replaces it with nothing?

Vegan vs. Recommended healthy diet

A question worth asking is how much water is saved consuming a "healthy 2000-calorie diet, for vegan and meat-eater. Recommended consumption is an equivalent for 3 cups of dairy and 5.5 oz of meat.

Non-vegan:

• Milk: 3 cups * 240 mL per cup * 1g/mL * / 1000 g per kg * 1020 L/kg: 734 L
• Meat: 5.5 oz * 28.34 grams per oz / 1000 g per kg * ((4325+5988+15415) / 3): 1336 L (the (4325+5988+15415) / 3 is the average L/kg for various meats, as above)

Total: 2070 L, or 546 US gallons

Plant milk (I'll assume home-made almonds (0.75 cups for equivalent nutrition, 20:80 almond:water) and beans (1.5 cups for equivalent nutrition as meat, so 0.5 cups dry + 2.5 cups water to cook)):

• Milk-equivalent: 0.75 * 130 (grams per cup) / 1000 * 9063 L/kg + 0.75*4: 886 L
• Meat-equivalent: 0.5 * 100 (grams per cup) / 1000 * 3375 L/kg + 2.5 * 240 / 1000: 169 L

Total: 1055 L, or 278 US gallons

A difference of about 268 US gallons. Using similar calculations for the Standard American Diet" (about 6.5 oz of meat vs 5.5 oz), we'd get a difference of roughly 316 US gallons.

Specific vegan diet

Replacing it all with cereals like oats, it'd be something like (assuming about 1.5 cups per day): 1.5 cups * 90 grams / cup / 1000 * 1644 L / kg + 1: 221 L, or 58 US gallons, which gets closer to the figure.

Summary

It depends on what the vegan eats, but the figure of 600 gallons is possibly for an average American who removes non-vegan foods from their diet entirely, without replacing them, or replacing them all with specific foods that are not water-intensive, which is probably unrealistic for an average person.

For vegans and meat-eaters eating equivalent, water-intensive, healthy diets, the vegan diet would save at least 250-300 gallons of water. With more care, the vegan diet could easily save up to 500 gallons.

Edit: relevant article for further analysis

This research article (not open source) appears to tackle the question more exactly: Mesfin M. Mekonnen & Julian Fulton (2018) The effect of diet changes and food loss reduction in reducing the water footprint of an average American, Water International, 43:6, 860-870, DOI: 10.1080/02508060.2018.1515571

It mentions a vegan diet having a water footprint 37% lower than the Standard American Diet (312 gallons vs. 493). It arrives at that result assuming an efficient system with no waste (my interpretation). Under those conditions, the diet would save 686 L (about 181 gallons) of water.