Short Answer: As the comments mention, this is a hot topic for pseudoscientists to use as a selling point for their supplements. There is limited evidence that dietary acid intake significantly alters bone resorption, especially in the setting of normal renal function. This paper should answer all of your questions (emphasis mine):
Frassetto, Banerjee, Powe, Sebastian. Acid Balance, Dietary Acid Load, and Bone Effects—A Controversial Subject. Nutrients. 2018. 10(4),517.
Abstract: Modern Western diets, with higher contents of animal compared to fruits and vegetable products, have a greater content of acid precursors vs. base precursors, which results in a net acid load to the body. To prevent inexorable accumulation of acid in the body and progressively increasing degrees of metabolic acidosis, the body has multiple systems to buffer and titrate acid, including bone which contains large quantities of alkaline salts of calcium. Both in vitro and in vivo studies in animals and humans suggest that bone base helps neutralize part of the dietary net acid load. This raises the question of whether decades of eating a high acid diet might contribute to the loss of bone mass in osteoporosis. If this idea is true, then additional alkali ingestion in the form of net base-producing foods or alkalinizing salts could potentially prevent this acid-related loss of bone. Presently, data exists that support both the proponents as well as the opponents of this hypothesis. Recent literature reviews have tended to support either one side or the other. Assuming that the data cited by both sides is correct, we suggest a way to reconcile the discordant findings. This overview will first discuss dietary acids and bases and the idea of changes in acid balance with increasing age, then review the evidence for and against the usefulness of alkali therapy as a treatment for osteoporosis, and finally suggest a way of reconciling these two opposing points of view.
This paper is open-access and I recommend you read it, but the conclusions summarize their findings nicely:
We suggest the way to put all of this information together is to agree...that bone alone cannot buffer the positive acid balance that one can calculate from measures of acid production and acid excretion. In addition, endogenous acid production goes up and down in an attempt to maintain systemic blood pH...
Diets high in acid precursors add to the body’s acid burden. For the majority of people eating typical western diets with acid loads of ≤1 mmol/kg, whose renal function and acid excretory ability is normal, dietary acid loads would not be a readily detectable factor in altering bone mineral density leading to the development of osteoporosis. Other factors such as age, gender, race, and immobility are quantitatively more major factors in determining bone mass and bone breakdown.
However, body retention of only 1 or 2 mEq of acid each day, barely detectable by current measurement techniques, buffered by muscle and kidney and titrated by skeletal base over decades, could potentially result in major depletion of bone mineral. Thus, we suggest that those older subjects with diminished renal function, decreased renal acid excretory ability, and lower buffering capacity due to lower muscle and/or bone mass, whose diets contain high net acid loads could potentially benefit the most from alkali therapies.
Long Answer: Although there's a wealth of pseudoscience on the internet relating to acidemia and bone composition, there is some truth to the idea that bone can be affected by pH. An excellent model to explore this concept is chronic kidney disease, as impaired renal function causes decreased H+ secretion, which results in a chronic metabolic acidosis:
Kopple, Kalantar-Zadeh, Mehrotra. Risks of chronic metabolic acidosis in patients with chronic kidney disease. Kidney International. 2005. 67(95),S21-27.
Severe chronic metabolic acidosis (i.e., the presence of excess hydrogen ions in blood) has two well-recognized major systemic consequences. First, metabolic acidosis, or acidemia, induces increased protein catabolism, decreased protein synthesis, and negative nitrogen and total body protein balance, which improve upon bicarbonate supplementation. Second, metabolic acidosis causes physicochemical dissolution of bone and cell-mediated bone resorption (inhibition of osteoblast and stimulation of osteoclast function).
Metabolic Acidosis and Bone Disease
Metabolic acidosis exerts multiple effects on bone. It causes the physicochemical dissolution of bone and cell-mediated bone resorption through the inhibition of osteoblast activity and stimulation of osteoclast function. These events are associated with a loss of calcium and phosphorus from the bone. Metabolic acidosis stimulates osteoblasts to release prostaglandins. This stimulates osteoclast function and inhibits osteoblast activity. Glucocorticoids can inhibit the production of prostaglandins by osteoblasts.
Metabolic acidosis is also associated with a decrease in the content of bone bicarbonate. Additionally, bicarbonate supplementation in postmenopausal women is associated with decreases in urinary calcium, phosphorus, and increased osteocalcin levels, indicating a beneficial effect. Figure 4 shows the mechanisms relating metabolic acidosis with bone disease3. These observations suggest that uncorrected metabolic acidosis has adverse consequences on bone anatomy and physiology.
Figure 4. The various mechanisms by which metabolic acidosis may contribute to bone disease. (From3).
From here, your questions can be answered:
Q: By consuming acidic foods, could one cause their blood to become acidic and therefore cause an acid+base reaction between their blood and bone?
A: Maybe. It is possible to induce a metabolic acidosis with diet:
Adeva, Souto. Diet-induced metabolic acidosis. Clinical Nutrition. 2011. 30(4),416-21.
The modern Western-type diet is deficient in fruits and vegetables and contains excessive animal products, generating the accumulation of non-metabolizable anions and a lifespan state of overlooked metabolic acidosis, whose magnitude increases progressively with aging due to the physiological decline in kidney function. In response to this state of diet-derived metabolic acidosis, the kidney implements compensating mechanisms aimed to restore the acid-base balance, such as the removal of the non-metabolizable anions, the conservation of citrate, and the enhancement of kidney ammoniagenesis and urinary excretion of ammonium ions. These adaptive processes lower the urine pH and induce an extensive change in urine composition, including hypocitraturia, hypercalciuria, and nitrogen and phosphate wasting.
And there is existing research on short-term effects of altered diet on blood and urine pH and composition, but the clinical relevance of these values is indeterminate:
Buclin, Cosma, Appenzeller, Jacquet, Décosterd, Biollaz, Burckhardt. Diet Acids and Alkalis Influence Calcium Retention in Bone. Osteoporosis International. 2001. 12(6),493-99.
The urine-acidifying properties of food constituents depend on their content of non-oxidizable acids or precursors. Acidifying constituents such as animal proteins may negatively affect calcium metabolism and accelerate bone resorption, thus representing an aggravating factor for osteoporosis...acid-forming diet increased urinary calcium excretion by 74% when compared with the base-forming diet (p50.0001), both at baseline and after the oral calcium load, and C-telopeptide excretion by 19% (p=0.01), suggesting a skeletal origin for the excess calcium output. This observation confirms that renally excreted acids derived from food influence calcium metabolism, and that alkalizing nutrients inhibit bone resorption. Further studies are needed to determine the clinical impact of dietary counseling for avoiding diet acids as a preventive measure against osteoporosis.
But best recent evidence does not support the idea that significant changes in bone composition could be induced by diet alone (see my "Short Answer" above).
Q: ...what salt would be produced?
There are many metabolic acids, but for simplicity, we'll consider H+. Bone bases are stored as alkaline salts (e.g. calcium carbonate, CaCO3, which is really Ca2+ and CO32-). So, if H+ comes into contact with calcium carbonate, you'll make carbonic acid (with H+ displacing Ca2+ to produce H2CO3 – you excrete the Ca2+, resulting in the hypercalcuria described above). Through the bicarbonate buffer system, that H2CO3 will be converted into carbon dioxide and water, which you excrete from your body via respiration.