Is it legal/permitted in the US public stock markets for an individual retail trader to do bid-ask spread scalping through algorithmic trading?
Do I need to follow some procedure before I can automate bid-ask scalping strategy?
In US public stock markets there is no difference between the actions individual retail traders are "permitted" to take and the actions institutional/corporate traders are "permitted" to take. The only difference is the cost of those actions. For example, if you become a Registered Market Maker on, say, the BATS stock exchange, you'll get some amazing rebates and reduced transaction prices; however, in order to qualify for Registered Market Maker status you have to maintain constant orders in the book for hundreds of equities at significant volumes. An individual retail trader is certainly permitted to do that, but it's probably too expensive.
Algorithmic trading is not the same as automated trading (algorithmic trading can be non-automated, and automated trading can be non-algorithmic), and both can be anywhere from low- to high-frequency. A low-frequency automated strategy is essentially indistinguishable from a person clicking their mouse several times per day, so: no, from a legal or regulatory perspective there is no special procedure an individual retail trader has to follow before s/he can automate a trading strategy. (Your broker, on the other hand, may have all sorts of hoops for you to jump through in order to use their automation platform.)
Last (but certainly not least) you will almost certainly lose money hand over fist attempting bid-ask scalping as an individual retail trader, whether your approach is algorithmic or not, automated or not. Why? Because the only way to succeed at bid-ask scalping is to (a) always be at/near the front of the queue when a price change occurs in your favor, and (b) always cancel your resting orders before they are executed when a price change occurs against you. Unless your algorithms are smarter than every other algorithm in the industry, an individual retail trader operating through a broker's trading platform cannot react quickly enough to succeed at either of those. You would have to eschew the broker and buy direct market access to even have a chance, and that's the point at which you're no longer a retail trader.